Let $f:[0,1] \to \mathbb{R}$ be continuous. The Stone–Weierstrass theorem tells us that there is a sequence $p_n$ of polynomials defined on $[0,1]$ such that $$\lim_{n \to +\infty} ||p_n - f||_{\infty} = 0$$
A consequence of this theorem is that
$$\lim_{n \to +\infty} \inf \{||p-f||_{\infty} : p \ \text{is a polynomial of degree} \leq n\} =0$$
This is the quantity I'm interested in. Namely, $$\epsilon(n) \stackrel{\rm def}{=}\inf \{||p-f||_{\infty} : p \ \text{is a polynomial of degree} \leq n\}$$
This roughly tells us the following: if we want to uniformly approximate $f$ by a polynomial of degree $\leq n$, then we should expect an error of at least $\epsilon(n)$.
Is there an explicit upper bound on $\epsilon(n)$ (e.g., something like $\epsilon(n) \leq \frac{15}{\log n})$? Is there an asymptotic bound on $\epsilon(n)$ (e.g., $\epsilon(n) = O(n^{-2})$?
The bounds should be as tight as possible.