Python, 40 bytes

f=lambda s,i=0:s[i:]and s+f(s[::-1],i+1)

A recursive function. Prepends the input string s to the function of the reverse until the counter i exceed the length of s.

Vim, 39, 34 keystrokes

:se ri
Y:s/./<C-r>"/g
<C-o>qqgJC<C-r>"<esc>gJ@qq@q

5 bytes saved thanks to @Lynn!

Here is a gif of it happening live: (Note that this gif is of a previous version since I haven't had time to re-record it yet).

And here is an explanation of how it works:

:se ri                  "Turn 'reverse indent' on.
Y                       "Yank this line
:s/./<C-r>"/g           "Replace every character on this line with the register
                        "We just yanked followed by a newline
<C-o>                   "Jump to our previous location
     qq                 "Start recording in register 'q'
       gJ               "Join these two lines
         C              "Delete this line, and enter insert mode
          <C-r>"<esc>   "Paste the line we just deleted backwards 
gJ                      "Join these two lines
  @q                    "Call macro 'q'. This will run until we hit the bottom of the buffer.
    q                   "Stop recording.
     @q                 "Start our recursive macro

On a side note, Y grabs an extra newline, which is usually an obnoxious feature. This is probably the first ever time that it has actually saved several bytes!

PHP, 54 52 bytes

(49 bytes, but do not work if string contains '0')

for(;($a=$argv[1])[$i++];)echo$i%2?$a:strrev($a);

(52 bytes)

<?=str_pad('',strlen($a=$argv[1])**2,$a.strrev($a));

(54 bytes)

for(;$i++<strlen($a=$argv[1]);)echo$i%2?$a:strrev($a);

Jelly, 4 3 bytes

,Ṛṁ

Try it online! or Verify all test cases.

Saved a byte thanks to @Maltysen.

Explanation

,Ṛṁ  Input: string S
 Ṛ    Reverse S
,     Join S with reverse of S. Makes a list [S, rev(S)]
  ṁ   Mold [S, rev(S)] to len(S) by repeating elements cyclically
      Return and print implicitly as a string

Brain-Flak, 418 bytes

This is my Brain-Flak masterpiece. It may not be well golfed but the challenge is the most difficult I have ever encountered.

Try it online!

(([])[()]){({}[()]<((({})<{({}[()]<(({}<>))<>>)}{}<>>)<{({}[()]<({}<>)<>([][()])({()<({}[()]<({}<({}<>)<>>)>)>}{}<>){({}[()]<({}<>)<>>)}{}<>>)}{}<>([]){{}({}<>)<>([])}{}<>([]){(({}[()])<{({}[()]<({}<({}<>)<>>)>)}{}<>([]){{}({}<>)<>([])}{}<>>)}{}>)>)}{}(({})){(({}[()])<({}<>)<>(({}))({<({}[()])><>({})<>}{})<>{}<>({}<({}<>)<>>)<>({}<>)({}<{(({}[()])<{({}[()]<({}<({}<>)<>>)>)}{}<>([]){{}({}<>)<>([])}{}<>>)}{}>)>)}{}{}

I am way to tired to explain it now, but I will explain it.

2sable, 3 bytes

Code:

gGÂ

Explanation:

g   # Get the length of the input
 G  # Do the following n - 1 times:
  Â # Bifurcate, which duplicates a and reverses the duplicate

Uses the CP-1252 encoding. Try it online!

Perl, 24 bytes

Includes +2 for -lp

Give input on STDIN:

rev.pl <<< Hello!

rev.pl:

#!/usr/bin/perl -lp
s%.%s/.?/chop/eg;$`%eg

Surprisingly this does not use the builtin reverse operator. That name is just soooo long, all solutions I could think of with reverse are at least 1 byte longer.

J, 13 8 bytes

Saved 5 bytes thanks to miles!

#;@$];|.

This is a 5-train with the following verbs:

# ;@$ ] ; |.

The inner fork is composed of ] (identity), ; (link), and |. (reverse). Observe:

   (| ; |.) 'Hello!'
+------+------+
|Hello!|!olleH|
+------+------+

The outer two verbs make the rest of the train. # is, in this case, the size of the argument, that is, the length. The verb linking these is ;@$, or ravel over reshape. Observe:

   # 'Hello!'
6
   6 $ (] ; |.) 'Hello!'
+------+------+------+------+------+------+
|Hello!|!olleH|Hello!|!olleH|Hello!|!olleH|
+------+------+------+------+------+------+
   ; 6 $ (] ; |.) 'Hello!'
Hello!!olleHHello!!olleHHello!!olleH
   6 ;@$ (] ; |.) 'Hello!'
Hello!!olleHHello!!olleHHello!!olleH
   (# ;@$ (] ; |.)) 'Hello!'
Hello!!olleHHello!!olleHHello!!olleH
   (# ;@$ ] ; |.) 'Hello!'
Hello!!olleHHello!!olleHHello!!olleH
   (#;@$];|.) 'Hello!'
Hello!!olleHHello!!olleHHello!!olleH

Old solution.

[:,|.^:(i.@#)

Simple enough. |. is reverse, and ^: is power conjunction, which repeats it's left verb (right hand) # of times. When the right argument is a verb, that verb is called on the argument. The right verb in this case is range from zero (i.) to the length (#). When raised to an array, the intermediate results are kept. All that needs to be done is to flatten the array with ,.

Intermediate results

   (i.@#) 'Hello!'
0 1 2 3 4 5
   |.^:0 1 2 3 4 5 'Hello!'
Hello!
!olleH
Hello!
!olleH
Hello!
!olleH
   |.^:(i.@#) 'Hello!'
Hello!
!olleH
Hello!
!olleH
Hello!
!olleH
   ([:,|.^:(i.@#)) 'Hello!'
Hello!!olleHHello!!olleHHello!!olleH

Minkolang, 17 bytes:

$oId$z$Dz[rz[O]].

Try it here!

Explanation

$o                   Read in whole input as characters
  Id                 Push the length of stack and duplicate
    $z               Pop top of stack and store in register (z)
      $D             Pop top of stack (n) and duplicate whole stack n-1 times
        z[     ]     z times, do the following:
          r          Reverse the stack
           z[O]      z times, pop the top of stack and output as character
                .    Stop.

Pip, 11 10 bytes

L#aORVRV:a

Try it online!

Explanation:

            a is first cmdline argument (implicit)
L#a         Loop len(a) times:
      RV:a   Reverse a and assign back to a
   ORV       Output the reverse of a (since it needs to go forward first then backward)

Ruby, 39 bytes

->(s){s.reverse!.gsub(/./){s.reverse!}}

I suck at Ruby. Golfing help is appreciated.

Ruby is a really nice language for this because of .reverse!

Explanation

I was hoping it would be someting simple like:

s.gsub(/./){s.reverse!}

but because of boilerplate/challenge restriction it's longer.

What s.reverse! is very useful. s.reverse! is basically s = s.reverse!, meaning it also mutates s.

What each section of the program does is described below:

->(s){             # Lambda with argument s
      s.reverse!   # Reverse `s` see above for details
      .gsub(/./)   # Replace every character with...
      {s.reverse!} # the input reversed!

The thing about s.reverse! that is great is that everytime it is evaluated the string get's flipped. So as it replaces the string. s is modified!

JavaScript (ES 6), 59 50 bytes

9 Bytes thanks to Hedi and Huntro.

f=(s,n=1)=>s[n]?s+f([...s].reverse().join``,n+1):s

recursive function.

Reversing the string takes almost half of the size (25 22 bytes!) ...
Why isn´t there a native way for that?

Java, 127 111 88 bytes

(s,r)->{for(int i=0;i++<s.length();)r+=i%2<1?new StringBuffer(s).reverse():s;return r;};

Ungolfed test program

    public static void main(String[] args) {
    BiFunction<String, String, String> func = (s, r) -> {
        for (int i = 0; i++ < s.length();) {
            r += i % 2 < 1 ? new StringBuffer(s).reverse() : s;
        }
        return r;
    };
    System.out.println(func.apply("Hello!", ""));
}

Haskell, 40 36 32 Bytes

m s=take(length s^2)$cycle$s++reverse s

Example:

*Main> m "Hello!"
"Hello!!olleHHello!!olleHHello!!olleH"

Even shorter (credit to Damien):

q s=zip(s>>[s,reverse s])s>>=fst

s>>[s,reverse s] cycles ["abc","cba",...] which is zipped to correct size and concatMap'ped with fst

PHP, 54 bytes

while($i++<strlen($s=$argv[1])){echo$s;$s=strrev($s);}

There are a lot of ways to do that, in almost every language I guess.

These two are my favourites:

an evil eval (62 bytes)

eval(str_repeat('echo$s;$s=strrev($s);',strlen($s=$argv[1])));

and a substr solution (69 bytes)

<?=substr(str_repeat(($s=$argv[1]).strrev($s),$n=strlen($s)),-$n*$n);

and while we´re at unsensible coding, take these 61:

$s=$argv[1];L:echo$s;$s=strrev($s);if(++$i<strlen($s))goto L;

Brachylog, 16 bytes

l-I,?r:?r:Ij@2hc

Try it online!

Explanation

l-I,                I = length(Input) - 1
    ?r              Reverse the Input
      :?r           The list [Input, Reverse of the Input]
         :Ij        Append I times that list to itself
            @2h     Split in half and take the first part
               c    Concatenate into a string

Retina, 27 bytes

Byte count assumes ISO 8859-1 encoding.

.
±$_
O$^`(?(\G)[^±]|±)

±

Try it online!

Explanation

.
±$_

Replace each character with the entire input, preceded by ± to separate copies.

O$^`(?(\G)[^±]|±)

The regex matches the characters in every other copy, by either matching a ± that is not adjacent to the previous match, or a non-± that is. These characters are reversed. Since each copy is the same width that is identical to reversing each copy individually.

±

Remove the separators.

CJam, 10 bytes

l_,({_W%}*

Try it online!

Straw, 10 bytes

<:(:>"),*&

Try it online!

<:(:>"),*&
<          Take the input       (Stack: ["", input])
 :         Duplicate            (Stack: ["", input, input])
  (        Start a string
   :       Duplicate
    >      Output
     "     Reverse
      )    End the string       (Stack: ["", input, input, ':>"'])
       ,   Swap                 (Stack: ["", input, ':>"', input])
        *  Unary multiplication (Stack: ["", input, ':>"'*length of input])
         & Evaluate             (Stack: ["", input])

PowerShell v2+, 57 bytes

param($a)-join(1..($x=$a.length)|%{($a[$x..0],$a)[$_%2]})

No real clean way to get string lengths or reverse 'em, so this is pretty lengthy.

Takes input $a, loops from 1 to $a.length (stored in $x for use later). Each iteration we use a pseudo-ternary to index into an array of either $a or $a[$x..0] (i.e., reversed), based on whether our input number is odd/even [$_%2]. These are all encapsulated in parens and -joined together to form a single string. That's left on the pipeline, and output is implicit.

PS C:\Tools\Scripts\golfing> .\stringgnirts.ps1 'TimmyD'
TimmyDDymmiTTimmyDDymmiTTimmyDDymmiT

Dyalog APL, 8 bytes

∊≢⍴⊂,⊂∘⌽

∊ flatten

≢ the tally

⍴ cyclically reshaping

⊂ the enclosed argument

, followed by

⊂∘⌽ the enclosed reversed argument

TryAPL online!

-1 byte thanks to jimmy23013.

Java, 151 bytes

public static void r(String s){String t = new StringBuffer(s).reverse().toString();for(int i=0;i<s.length();i++){System.out.print(((i%2==1)?t:s));}}

}

Ungolfed:

public static void r(String s) {
    String t = new StringBuffer(s).reverse().toString();
    for(int i = 0; i < s.length();i++) {
        System.out.print(((i % 2 == 1) ? t : s));
    }
}

C#, 94 bytes

using System.Linq;string R(string n)=>string.Concat(n.SelectMany((c,i)=>1>i%2?n:n.Reverse()));

76 bytes for the method + 18 bytes for LINQ import.

How it works:

using System.Linq; // Required for LINQ extension methods.

string R(string n) => 
    string.Concat( // Concatenate the following chars into a single string
        n.SelectMany( // Enumerate each char in n, flattening the returned IEnumerable<char>'s into a single IEnumerable<char>
            /*IEnumerable<char> Lambda*/(/*char*/ c, /*int*/ i) => // i = index in n
                1 > i % 2 // Check if i is even or odd
                    ? n // if i is even, add n to the concat
                    : n.Reverse() // else reverse n and concat that
        )
    )
;

Vim + coreutils, 32 keystrokes

You can never have too many Vim answers.

qqYv:!rev
Pjq@=len(@")
@q2dkv{gJ

Explanation

qq               " Start recording macro
Y                " Yank (copy) line
v:!rev<CR>       " Reverse line with coreutils rev command
Pj               " Paste yanked line above this line
q                " Stop recording
@=len(@")<CR>@q  " Playback macro once for each character
2dk              " Delete last 3 lines
v{gJ             " Join lines

Ruby, 43 bytes

f=->s,i=s.size{$><<s
i<2||f[s.reverse,i-1]}

See it on eval.in: https://eval.in/642616

Actually, 12 bytes

;;l)Rkp;)αHΣ

Try it online!

Explanation:

;;l)Rkp;)αHΣ
;;l           dupe string twice, use one copy to get length (call it L)
   )          move length to bottom of stack
    R         reverse one copy of string
     k        pop entire stack and push as list
      p       pop L from list
       ;)     dupe L, move one copy to bottom
         α    repeat each element (the forward and reversed string) of the list L times
          H   first L elements in list (trim off excess elements)
           Σ  concatenate

Groovy, 51 bytes

def d(s){s.size().times{print(it%2?s.reverse():s)}}

C#, 137 bytes

void S(string i){Console.WriteLine(string.Join("",i.Select((c,n)=>(n%2==0)?i:i.Reverse()).Select(x=>new string(x.ToArray())).ToList()));}

Ungolfed:

void S(string i)
{
    Console.WriteLine(string.Join("",
    i.Select((c, n) => (n%2 == 0) ? i : i.Reverse()).Select(x => new 
    string(x.ToArray())).ToList()));
}

R, 53 bytes

Assumes that the input is space- or newline-separated for each character.

cat(rep(c(i<-scan(,""),rev(i)),l=length(i)^2),sep="")

Some test cases:

> cat(rep(c(i<-scan(,""),rev(i)),len=length(i)^2),sep="")
1: h e l l o !
7: 
Read 6 items
hello!!ollehhello!!ollehhello!!olleh

> cat(rep(c(i<-scan(,""),rev(i)),l=length(i)^2),sep="")
1: a
2: 
Read 1 item
a

> cat(rep(c(i<-scan(,""),rev(i)),l=length(i)^2),sep="")
1: a b c d
5: 
Read 4 items
abcddcbaabcddcba

> cat(rep(c(i<-scan(,""),rev(i)),l=length(i)^2),sep="")
1: O K !
4: 
Read 3 items
OK!!KOOK!

> cat(rep(c(i<-scan(,""),rev(i)),l=length(i)^2),sep="")
1: 4 8 1 5 1 6 2 3 4 2
11: 
Read 10 items
4815162342243261518448151623422432615184481516234224326151844815162342243261518448151623422432615184

> cat(rep(c(i<-scan(,""),rev(i)),l=length(i)^2),sep="")
1: P P C G
5:    
Read 4 items
PPCGGCPPPPCGGCPP

> cat(rep(c(i<-scan(,""),rev(i)),l=length(i)^2),sep="")
1: 4 2
3: 
Read 2 items
4224

Octave, 39 bytes

@(x)[x'+0*x;flipud(x'+0*x),''](1:end/2)

f('Hello!')
ans = Hello!!olleHHello!!olleHHello!!olleH

Explanation:

@(x)            % Take x as input, inside apostrophes 'Hello!'
x'+0*x          % Create a mesh of the ASCII-code of the input letters
                % For input `bcd` this will be:
                %    98    98    98
                %    99    99    99
                %   100   100   100
;flipud(x'+0*x) % Concatenate vertically to create:
                %    98    98    98
                %    99    99    99
                %   100   100   100
                %   100   100   100
                %    99    99    99
                %    98    98    98
___,'']         % Short cut to convert ASCII-code to characters
(1:end/2)       % Display the first half of this array of letters, as a
                % horizontal string