Can a word be contracted twice (e.g. "I'ven't")?

This is not the highest register, but you may hear it in speech. Native speakers tend to slur words together and leave out sounds even if they wouldn't write that way.

Double contractions are not used in writing. They may be grammatically correct, but a professor would not allow you to use them in an essay. Typically, even single contractions are avoided in formal writing.

The example you give is not done in American English. You can't contract non-auxiliary "have". "I've not a clue if this is possible" is also not grammatical in American English. It may be in British English, though.

I copied this from a comment I left below, because I think it clarifies what I'm trying to say:

I've" is a fine contraction, just in American English you can only use it to replace "I have" when "have" is used as an auxiliary verb (e.g. in conjunction with a past participle). "I've been there" is OK. "I've a dog" is not. In the example "I'ven't a clue"—"I haven't a clue", the verb "have" is not auxiliary, so it can't be contracted with the pronoun "I".

"Fish 'n' chips" and similar phrases with "'n'" technically have a double-contracted "and."

I don't think you can get away with "I'ven't" in writing either. However, I think I've read "'tweren't," "'twouldn't," and "'twasn't" before. I'm guessing, though, that double contractions like that are never technically correct. Colloquially speaking, if you're understandable, anything goes.