{%}+")1$~":x`+0x

{%}+")1$~":n`+0

$ echo -n 3 | golfscript quining.gs
{3 % ")1$~"}0)1$~
$ echo -n 3 | golfscript quining.gs | golfscript
{3 % ")1$~"}1)1$~
$ echo -n 3 | golfscript quining.gs | golfscript | golfscript
{3 % ")1$~"}2)1$~
$ echo -n 3 | golfscript quining.gs | golfscript | golfscript | golfscript
{3 % ")1$~"}0)1$~

# (implicit) Read from STDIN.
{%}+             # Concatenate the read input with the block.
                 # For example, "3"{%}+ -> {3 %}.
    ")1$~":x     # Push that string and save it in the the variable `x'.
            `+   # Inspect the string and concatenate it with the block.
                 # For example, {3 %}'")1$~"'+ -> {3 % ")1$~"}
              0x # Push a 0 and the string stored in `x'.

{          }      # Define a block.
 3 %              # Take the integer on the stack modulo 3.
     ")1$~"       # Push that string.
            0)    # Push 0 and increment. Pushes 1.
              1$~ # Copy the block from the bottom of the stack and execute it.