What I really like and admire about Euler is how masterfully he handled infinities and infinitesimals to arrive at correct conclusions (the vast majority of times anyway) even though analysis hadn’t been made rigorous yet (by Cauchy and Weierstrass and friends), so some of what he did was pure magic and took a lot of good intuition.

An example of this is when, in solving the then notorious Basel problem, he factors trig functions into infinite products of (x +- k*pi) terms just by analogy with root factorization in finite polynomials.