latest registration for contact
Apparently this information (more likely a foreign key) is held somewhere in the contact record, since it's manifest in the Contacts UI, but it's not exposed by the Contacts API. I can retrieve it using the EventRegistrations API with the ContactId and throwing away all but the first record returned, but this takes a lot of wallclock time (suggesting that the server is working pretty hard to collect it).
No new API is needed, but the Contacts API could just include the most recent EventRegistration ID so one could navigate to it, or through it to the Event record as appropriate.
I note in passing that there is a numeric Custom Field 'Registered for specific event'/'RegistredForEvent' but it's not returned by my Contacts call.