Why not "I can give you ANOTHER chance"? Is this a wrong English sentence? Doesn't it have similar meaning to "I can give you one more chance"? Of course "one more" is a more literal translation of "jeszcze jeden", but isn't "another" almost the same as "one more"? In ABBYY Lingvo the Russian translation of "another" is "ещё один" which is EXACTLY the Polish "jeszcze jeden".
There is one chance only. That is the concept of "having a chance",
eg. "You only have one chance to make a good first impression". In very specific circumstances, you may be given another opportunity, the second chance, to prove yourself...
The third, forth, and seventh are not chances; they are... attempts.