How many essay competitions/supercurriculars can I join? Any recs?

I'm looking to study archaeology/classics, and I'm going be to applying to Cambridge as one of my choices. I know that for Cambridge I can only do one of the Fitzwilliam essay competitions, but there's a competition on colonialism from Oxford (the Rex Nettleford Essay Prize) that actually looks really interesting, as well as the Mary Renault Prize. Can I do both as they're different universities or can I still only choose one?

It'd also be great if someone could let me know what extra curriculars would be good for these kind of subjects, I'm already learning Greek outside of college (at a basic level currently though) and doing some reading (Tom Holland's Persian Fire, scholarship for A level Classics) but any other recommendations would be most welcome. I do quite a lot of other hobbies, but they're not particularly related to the course (music etc).

Thank you mysterious all knowing peoples of the internet!