1. Python 3 (8 bytes)

print(1)

This program prints 1 in Python 3.

Starting this off with Python 3 because I know it's good for polyglots and can be taken in a number of different directions (also, I wanted to ensure that the first answer was in a relatively normal language, rather than an absurd esolang that's hard to polyglot with).

4. ><> (29 bytes)

#>>>>>>>>v;n4
print(1)#>3N.i2

This program prints 1 in Python 3, 2 in Vim, 3 in Minkolang v0.15 and 4 in ><>

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Code ran

#             - change direction to left
            4 - add 4 to stack
           n  - print as a number
          ;   - end the program

Yet another 2D language.

Has no effect on Minkolang as it adds characters after the direction changes, gets ignored by Vim for some reason. # is a comment in Python so no change their either.

6. SMBF, 45 bytes

#v<++++<;n4
#>3N.
print('1'if 1/2else'5')
#i2

Try it online

This program prints 1 in Python 3, 2 in Vim, 3 in Minkolang v0.15, 4 in ><>, and 5 in Python 2.

SMBF (aka Self-modifying Brainfuck) uses <++++<>.. The pointer is moved left (to the last character of the source code), and the cell is incremented four times then printed.

8. Retina, 70 bytes

#v;7||"<+0+0+0+<;n4
#>3N.
#|\w*
#8
#|

#M`
print('1'if 1/2else'5')
#i2

Try it online

This program prints 1 in Python 3, 2 in V, 3 in Minkolang v0.15, 4 in ><>, 5 in Python 2, 6 in SMBF, and 7 in Japt.

Screw creating hyperlinks to interpreters. That's going to take way too much time. It's up to the poster and anyone creating new answers to test it yourself.

9. Perl, 84 bytes

#v;7||"<+0+0+0+<;n4
#>3N.
#|\w*
#8
#|

#M`
print(None and 9or 1/2and 1or 5)
#jd5ki2

There's a literal ESC character between the j and d on the last line, but my browser doesn't want to put it into the input box. (Perhaps someone with a different browser could edit it in?)

This prints 1 in Python 3, 2 in Vim (tested locally, but here's a link for the very similar language V), 3 in Minkolang, 4 in <><, 5 in Python 2, 6 in SMBF, 7 in Japt, 8 in Retina, and 9 in Perl.

Let's get some more exoteric languages in, via abusing arithmetic that works differently in different languages. (None is falsey in Python but truthy in Perl, and and/or chains work the same way in both languages.)

Apart from Python, I also had to change the vim code. Instead of making it into a series of no-ops, I just let it insert junk, then deleted the junk again at the end.

2. Vim (11 bytes)

print(1)#i2

This program prints 1 in Python 3, and 2 in Vim.

Just to get the ball rolling and to throw my favorite language into the mix early on. :)

It's a very straightforward answer.

print(1)#

just so happens to be a NOP in vim. (lucky for me) Then i2 enters insert mode and inserts a '2'. You can try vim online here

Of course, in python

print(1)

prints '1', and

#2

is a comment.

3. Minkolang v0.15 (26 bytes)

#>>>>>>>>v
print(1)#>3N.i2

This program prints 1 in Python 3, 2 in Vim, and 3 in Minkolang v0.15

I hope I don't mess things up by introducing a 2d language

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Explanation

#                     stops program from moving through time (really does nothing)
 >>>>>>>>             I can't use a space because then the program will move through time
         v            go down
         >            go right
          3N.         Outputs 3 and end program
                      Anything afterward is ignored since program has ended

Vim somehow ignores Minkolang, so that's good

And there really wasn't a problem with Python since it ignores the comments #

Next...

For the next language, I suggest something like ><> since # acts as a reflector (so that the direction will change to left and it will wrap to all the way in the right) so you can add code that can be ignored by other languages

5. Python 2 (35 bytes)

#3N.;n4
print('1'if 1/2else'5')
#i2

This program prints 1 in Python 3, 2 in Vim, 3 in Minkolang v0.15, 4 in ><> and 5 in Python 2.

Try It Online beta!

In Python 2, 1/2 is 0, which is a falsy value, which makes Python print 5. In Python 3, 1/2 is 0.5, which is a truthy value, which makes Python print 1.

7. Japt (50 bytes)

#v;7||"<++++<;n4
#>3N.
print('1'if 1/2else'5')
#i2

Man, it was fun to fit Japt in there. Hopefully it didn't kill anyone's plans...

Test it online!

This program prints 1 in Python 3, 2 in V, 3 in Minkolang v0.15, 4 in ><>, 5 in Python 2, and 6 in SMBF.

How this works: #v; takes the char code of v, or 118. Then 7|| returns the logical OR of 7 and the rest of the code, which is wrapped in a string with " to avoid any syntax errors. The result, 7, is automatically sent to STDOUT.

For future polyglotters, " can be changed to ` at no penalty to the Japt program (though I'm not sure about the others).

10. Befunge, 95 bytes

#v02^0;7||"<+0+0+0+<;n4
#v0#@00
#>3N.
#|\w*
#8
#|
#M`
print(None and 9or 1/2and 1or 5)
#jd5ki2

There is a literal ESC character between j and d on the last line (grr, @ais523). It is not included in this code. To get the actual code, please go to the Try it online link.

This prints 1 in Python 3, 2 in Vim, 3 in Minkolang, 4 in <><, 5 in Python 2, 6 in SMBF, 7 in Japt, 8 in Retina, 9 in Perl, and 10 in Befunge.

This code shares * with Retina and . with Minkolang and SMBF.

Try it online

Explanation

Actual program

#v02^
    @
    .
    *



    t
    5
#v02^

The last line was written for clarity (Befunge playground is cyclic.)

#

Trampoline, skips v

02^

Push 0 and then 2 in stack and go up.

5t*.@

Push 5, no-op, multiply two elements in stack (2 and 5), print, end program.