What are some interesting examples of axioms?

Axioms are essentially premises that are "self-evident." It's kind of hard to explain, you would use an axiom to base your other deductive reasoning/logic. One axiom I can think of is "I exist" like Descartes "I think, therefore I am." He uses his existence to prove that there is an external world. Another philosopher, I can't remember his name, uses "I am being appeared to green leafly" as an axiom because it cannot be proven otherwise that he is experiencing a "green leaf". Here's the wikipedia link, I hope you find it helpful.