The following English verb forms can appear as auxiliary verbs.[1]:p.19;[2] Note that some of these forms can also be used as main verbs. The main criterion used here for whether something is an auxiliary verb is whether it participates in subject-auxiliary inversion.
The contracted forms can be stacked, e.g. I'd've told her to leave, or She'll've left already by the time you get there.
Dare, need, used (to), and ought (to) are sometimes[1]:p.19 considered auxiliaries, but they do not permit subject-auxiliary inversion in many dialects. When they are auxiliaries, they permit sentences such as Dare you go? (with non-auxiliary equivalent Would you dare to go?), Need you say this? (Do you need to say this?), Used we to go?, and Ought we to go through with this? ((Do we have to / Should we) go through with this?). When dare and need are used as auxiliaries, that use is always in either the interrogative or the negative (He dare not go; He need not go), neither the auxiliary nor the main verb conjugates for the third person singular, and the particle to is not included before the bare infinitive. However, Warner[2]:p.8 rejects ought and used as auxiliaries because the subsequent infinitive form includes the particle to.
Palmer[1]:p.40 gives arguments for also including better or 'd better (or had better) as auxiliaries (as in the unconjugated He had better go, He had better not go, Had he better not go?). But Warner[2]:p.3 does not include it.