I once ran gzip and infinite number of times on a text file. I was surprised at the result.
Without actually checking, the result is going to be that the output size increases slightly over time.
The size definitely gets bigger with each iteration:
$ echo text >0.txt
$ for i in {0..9}; do
gzip <$i.txt >$((i + 1)).txt
done
$ ls | sort -n | xargs -n1 wc -c
5 0.txt
25 1.txt
46 2.txt
69 3.txt
82 4.txt
105 5.txt
120 6.txt
143 7.txt
161 8.txt
184 9.txt
207 10.txt