Third time the charm
Wumpus, 6 bytes

{~)
@O


Try it online!
Try it doubled!
Try it tripled!

Prints 1 and 3.

Explanation

I found a ton of 6-byte solutions by brute force search, but none for 5 bytes. That doesn't necessarily mean there aren't any at 5 bytes but they'd probably use weird characters or something.

I ended up picking this solution because it doesn't print any leading zeros (most of them do) and it has some interesting control flow. Let's start with the single program:



So the executed code is:

{~)O@

{   Turn the IP left by 60°.
~   Swap two implicit zeros on the stack, does nothing.
)   Increment the top zero to 1.
O   Print it.
@   Terminate the program.


Easy enough. Now the doubled program. Since the first line gets appended onto the second line, the grid extends to width 5 (and height 3) which changes the control flow significantly:



The IP goes around that loop exactly once, so the executed code is:

{~){~)){~O@

{~) As before, we end up with a 1 on top of the stack.
{   Turn left by 60° again.
~   Swap the 1 with the 0 underneath.
))  Increment the zero to 2.
{   Turn left by 60° again.
~   Swap the 2 with the 1 underneath.
O   Print the 1.
@   Terminate the program.


Finally, the tripled program is quite similar to the doubled one, but we get a couple more important commands onto that third line:



So the executed code is:

{~){~)){~~)O@

{~){~)){~
    As before. We end up with a 1 on top of the stack and a 2 underneath.
~   Swap the 1 with the 2 underneath.
)   Increment the 2 to a 3.
O   Print the 3.
@   Terminate the program.