Twenty years ago, on my graduation from law school, I would have liked to have had the chance to join the partnership, but the substantial investment required would have made such a move impossible.
2 events in the past liking and the opportunity to partner.
(A) I would have liked to have had the chance
Opportunity presents first and liking of the opportunity happens next hence " had is needed for opportunity.
Also this is hypothetical situation hence would is used in the past tense.
(B) I would like to have the chance
This still talks about a hypothetical present tense.
(C) I like to have the chance
This still talks about a present tense.
(D) I will like to have had the chance
WOW future(will), have (present) and had ( past) all used in simple tense form.
(E) I would like to have had the chance
would have is correct and have is missing here.