Batch, 46 bytes

@echo Press any key to continue...
@pause>nul

Because pause's output contains a space before each ..

HTML + JavaScript (ES6), 36 + 25 = 41 bytes

You can't really exit a JavaScript program, so clearing the webpage is the best I can think of.

onkeyup=_=>a.innerHTML=''

<a id=a>Press any key to continue...

Bash, 46 43 42 bytes

Saved 1 byte thanks to @DigitalTrauma

read -rn1 -p"Press any key to continue..."

Uses the read built-in. -r makes sure it doesn't allow the user to input escapes. -n 1 allows just one character. -p is the prompt

TI-Basic, 56 bytes

Basically, it loops until a key is pressed. Too bad lowercase letters are two bytes each in TI-Basic...

Disp "Press any key to continue...
Repeat getKey
End

Ruby (2.3), 52 55 54 53 bytes

Now 53 bytes by removing parentheses.

Note: Tested on Windows, might not work if there is no pause command.

puts"Press any key to continue...";system "pause>nul"

Explanation

Puts the required text:

puts"Press any key to continue..."

Run the Batch pause command and pipe output to nul:

system "pause>nul"

Octave / MATLAB, 42 bytes

disp('Press any key to continue...');pause

Bash 48 bytes

read -rn 1 -s -p "Press any key to continue... "

Pascal, 75 bytes

This was tested with the Free Pascal Compiler, version 3.0.0.
It may work with TurboPascal 7 or newer.

program _;uses Crt;begin write('Press any key to continue...');readkey;end.

Sadly, I can't replace readkey with readln since the challenge requires that any key be accepted.

I've tested this code on http://rextester.com/l/pascal_online_compiler, with and without supplying an input.
As expected, the program is terminated after 10s, since it sits waiting for a keypress that never happens.

Mathematica, 62 bytes

EventHandler["Press any key to continue...","KeyDown":>Exit[]]

Explanation

EventHandler["Press any key to continue...","KeyDown":>Exit[]]
EventHandler[                                                ]  (* Create a new EventHandler *)
             "Press any key to continue..."                     (* Print the dialog text *)
                                            "KeyDown":>Exit[]   (* When a key is pressed down, exit *)

Haskell, 51 bytes

main=putStr "Press any key to continue...">>getLine

Try it online!

Perl 5, 79 bytes

system "stty cbreak";$|=1;print "Press any key to continue...";read(STDIN,$a,1)

used as:

perl -e 'system "stty cbreak";$|=1;print "Press any key to continue...";read(STDIN,$a,1)'

No prizes of course. I'm sure some perl person will have a better way.

(89 bytes if the interpreter invocation as well needs to be included in the count)

QBasic (QB64), 37 (42?) bytes

Unfortunately, QBasic's built-in end-of-program prompt doesn't have the ellipsis, so we'll have to print it ourselves:

?"Press any key to continue..."
SLEEP

(SLEEP without an argument waits until a keypress.)

This code does what the question literally asks for, but it doesn't seem like it fits the spirit of the question because, of course, QBasic then displays "Press any key to continue" and waits for a keypress before returning to the IDE. Here's one that goes straight to the IDE, for +5 bytes:

?"Press any key to continue..."
SLEEP
STOP

STOP is a debugging statement. In regular QBasic, it sets a breakpoint: execution halts and we return to the IDE, but execution can be resumed again with F5. It's unclear whether that would count as the program "exiting." However, I'm using the QB64 emulator, which can't do breakpoints. Upon encountering STOP, it simply halts--returning straight to the IDE without the redundant "Press any key" message.

Processing, 89 81 bytes

void setup(){print("Press any key to continue...");}void draw(){if(key>0)exit();}

Explanation (outdated)

void setup(){print("Press any key to continue...");}

This is required since I am using more than one function in my program. Anything inside setup() gets called, in this case the string "Press any key to continue...".

if(key>0)exit();

Checks if key (key will always contain the int value of the last key pressed) is more than 0 (ie not a null byte). If the condition is satisfied, then the program exits.

void draw(){}

draw() ensures that that the program will always keep looking for a key instead of stopping once the program starts.

(That feel when a builtin in a Java-like language is still verbose...)