Ruby: 31 21 characters
(This is a rewrite of AbhijitAbhijit's Python solutionPython solution. If you like the base idea, upvote his answer, like I did.)
open($0,?a)<<"+1";p 1
Sample run:
bash-4.2$ ruby increment.rb
1
bash-4.2$ ruby increment.rb
2
bash-4.2$ ruby increment.rb
3