Approximate degree of $\textrm{AC}^0$
A paper by Mark Bun and Justin Thaler has been posted on ECCC very recently (mid-March 2017) that precisely answers this question: "A Nearly Optimal Lower Bound on the Approximate Degree of AC0"

They claim that for any $\delta &gt; 0$, there exists a function $f$ in $\mathrm{AC}^0$ such that $\widetilde{\mathrm{deg}}_{1/3}(f) = \Omega(n^{1-\delta})$, nearly closing the gap with the trivial $O(n)$ upper bound. They achieve this with a general method to boost the approximate degree of a function with sublinear approximate degree, keeping the number of variables quasi-linear. From the abstract :


  Specifically, we show how to transform any Boolean function $f$ with approximate degree $d$ into a
  function $F$ on $O(n \cdot \mathrm{polylog}(n))$ variables with approximate degree at least $D = Ω(n^{1/3} · d^{2/3} )$. In
  particular, if $d = n^{1−Ω(1)}$, then $D$ is polynomially larger than $d$. Moreover, if $f$ is computed by a
  polynomial-size Boolean circuit of constant depth, then so is $F$.


That is the most recent update on the lower bound end of this problem, and it is quite a significant step forward. The Introduction and Application sections of the paper are also good sources of references for prior works and related problems.

Disclaimer: I haven't read the paper carefully yet.