What’s the infinitive of the verb I use when I say “I might go” or “May I come with you”?
I think in German it’s dürfen. Is there one in English? If not, why not?
May is a modal verb, so it is "defective" in that it does not have the usual infinitive with "to". Might is a past-tense form of may — though, as Wikipedia puts it, it has "acquired an independent, present tense meaning". A very similar thing happened in German, where möchte has a life of its own, and many native speakers don't realize that it started off as the subjunctive form of mögen (etymologically, that's the German equivalent of may, but it has drifted away to mean "to like" in contemporary German).
When using the words "may" and "might" as conditional helper verbs, I do not believe there is an infinitive that starts with "to", the way most other English verbs do. One way of rephrasing "I might go" or "I may go" is "It is possible that I will go", which indicates a helper verb, not an infinitive phrase.
"May I come with you?" is a different use of the word "may", which indicates asking for permission. The equivalent infinitive phrase to be used in the form "to [verb]" for that verb might be "to have permission". This is not the case in some other languages, in which case there is a infinitive that is conjugated. For example, in Spanish, the verb "poder" is often used to express the same sentiment as "can" and "may" in English, and is best translated as "to be able", although it's also used as "to have permission". "¿Puedo venir contigo?" can be translated as "Can I come with you?" (do I have the ability to?) or "May I come with you?" (do I have permission to come?)
There are some possible reasons for this; my best guess is that English is sometimes structured differently, and it uses helper verbs, which I don't think really exist in other languages (at least, those with which I'm familiar.)