What determines them?
Stats, right?
And how do you raise your pets tier, by increasing stats?
Also, how many tiers are there?
(Please don't say an infinite amount that'd make me and all my tier 1 pets feel like complete failures...)
There are 13 tiers! c: You gain tiers by increasing your strength, defence, health and speed stats to predetermined levels. Your level and intelligence stats don't count towards your tier. Level is only used to train your other stats, so its best to keep it as low as possible until you have to train it. link to tier guide
No worries! Let me know if you have any other questions. :D
Your tier 1 pets needn't worry too much. If you train them consistently, you can get to tier 5/6 in a year, and tier 8/9 in two years. Once you hit tier 13 the experience point costs go up much higher, so while it's the top tier now if they add another one, it'll take a good while for folks to reach it without massively boosting.
It's pretty rare for site events to have compulsory combats, and most of the time those are less than tier 4.