If you're using a shell such as Bash 4, then ** is essentially a recursive version of *, which will match any number of subdirectories.
This makes more sense if you add a file extension to your examples. To match log files immediately inside tmp, you would type:
/tmp/*.log
To match log files anywhere in any subdirectory of tmp, you would type:
/tmp/**/*.log
But testing with git version 1.6.0.4 and bash version 3.2.17(1)-release, it appears that git does not support ** globs at all. The most recent man page for gitignore doesn't mention **, either, so this is either (1) very new, (2) unsupported, or (3) somehow dependent on your system's implementation of globbing.
Also, there's something subtle going on in your examples. This expression:
tmp/*
...actually means "ignore any file inside a tmp directory, anywhere in the source tree, but don't ignore the tmp directories themselves". Under normal circumstances, you'd probably just write:
/tmp
...which would ignore a single top-level tmp directory. If you do need to keep the tmp directories around, while ignoring their contents, you should place an empty .gitignore file in each tmp directory to make sure that git actually creates the directory.