"No way". Use this when you:
---mean no chance/possibility --as in-- "No way, I'll never let you fail."
---are expressing disbelief --as in-- "OMG! You won the lottery? No way!"
"No Chance". Use this when you:
--- mean no possibility (in this sense it's interchangeable with 'no way') --as in-- "No, you have no chance of winning the lottery"
"No sh*t" (This is actually vulgar) Only use this with friends or people you hang around. It can be used:
---sacrastically, when you respond to something obvious. In this sense it means "DUHH" For example:
Friend: "OMG, did you know that the earth revolves around the sun?"
You: "No sh*t"
(No sh*t is not really used to say "no". It's just a negative response to something obvious. Or it can be used as a vulgar/slang way to say that you are serious about something. For example, "I just saw Bill Clinton having pizza with spongebob, no sh*t"--- meaning that you're serious.)
If you mean that you want other alternatives to 'no' by "Any other no " then you can say "nope, nah (more slang-y), never, in no way shape or form, not in this universe, etc.." But then again, it would depend on what you mean =)