"He would have never dreamed that he would get the grant or that he would have got the grant ? " which is correct? according to most grammars only the first is correct(future i the past) but I gather from sources and spoken language that the second alternative is also correct or is it just tolerated as many other incorrect utterances/expressions are?
Both would+infinitive and would have+past participle can be used to express the future in the past.
If you said "I believe that I will get the grant", that would backshift to "I believed that I would get the grant".
If you said "I believe that I will have got the grant", that would backshift to "I believed that I would have got the grant".
You could say either "I can't imagine that I will get the grant" or "I can't imagine that I will have got the grant", depending on the intended shade of meaning, and when talking about the past, the latter backshifts to "I couldn't imagine that I would have got the grant".
From Thomson & Martinet, A Practical English Grammar (1985), ch. 21:
The perfect conditional... is used... (c) as a past equivalent of the future perfect:
I hope he will have finished before we get back.
I hoped he would have finished before we got back.
(Note: BrE uses "I have got" and "I would have got", etc; AmE uses "I have gotten" and "I would have gotten", etc; both varieties use "I have got" as a colloquial equivalent to "I have" as a present tense main verb expressing possession.)