Welcome!
I'm wondering if theres other types of contests mainly. Theres a few situations that are likely to come up which are probably going to be either a race or a cat and mouse. However whilst I can get a lot out of those two alone. I'm wondering about my options. If theres a topic on this already feel free to link away.
There are, more in blogs and forums than directly from the book I think.
You can anthropomorphize almost anything - particularly when abstracted to game mechanics. Take the "FATE fractal" concept as an example. Characters have four defining things: aspects, skills, powers/stunts, and stress tracks. Other items in the game often have one or more of the same 'defining things'. Cities have aspects, so do scenes and even adventures. A chase may be abstracted to a stress track, a hazard (such as fire) may have all or nearly all of a character's defining attributes. I build organizations with aspects and, when under direct attack by the PCs, a stress track. They'll even have a few skills such as Resources or Contacts. You could easily extend that to armies or even nations should you want a campaign built around war between states.
In short, if you can state it as a character-like entity, you can use it in a contest with game mechanics.
Edit: A few links...
the Core of FATE Core discusses the basics of FATE and introduces the fractal while
Stress Consequences and the Fractal takes a deeper dive into stress and consequences.
It's On Fire talks about extending the character concept to hazards instead of just using an aspect and
a group on RPG.net discussed representing a wilderness obstacle as a character. I'm sure there are other links...(in)distictly remember a few discussions touching on the subject in these forums. Afraid I don't have links though.