Ananyan-Hochster's recent proof of Stillman's conjecture crucially uses the notion of strength of a polynomial. Inspired by this, we present two results which capture the idea that collections of polynomials of sufficiently high strength behave like regular sequences. Both of these results state that suitable limits (ultraproduct and inverse limit, respectively) of polynomial rings are themselves polynomial rings. I will discuss how the first result leads to a different proof of Stillman's conjecture. Time permitting, I will also discuss the inverse limit version and how it connects to recent work of Draisma on GL_infinity-noetherianity of polynomial functors.