I was watching Carl Bender's lecture 2 on YouTube and I stumbled across a 'fundamentally hard problem', namely the Schrodinger equation
$y''(x)+Q(x)y=0$ (eq.1)
Carl mentions that this is a 'fundamentally hard problem' because the most natural way to proceed with solving (eq.1) reduces your calculations to solving that same original problem! In other words, there is nothing you can do to solve (eq.1). He does this to motivate the power of asymptotic methods by sacrificing the notion of equal signs (=) and introducing a new notion (~).
My question goes beyond this, in particular, are there asymptotic problems which are fundamentally hard? If so, what notion from asymptotics must we sacrife in order to make progress?