Classes of boolean functions where reasonable lower bounds on approximate degree is unknown? Let $\underline{x}\triangleq x_1,\dots,x_n$. 
Given a Boolean function 
$f:\{0,1\}^n\to\{0,1\}$, 
we say that $p(\underline{x})\in\Bbb{R}[\underline{x}]$ is an $\epsilon$-approximation to $f$ 
if  for all $\underline{x}$, $|p(\underline{x})-f(\underline{x})|\leq\epsilon$.
$\newcommand{\apxdeg}[0]{{{\mathsf{deg}}}}$
Let $\mathsf{deg}_\epsilon(f)$ be the minimum degree of
multilinear real polynomial that $\epsilon$-approximates $f$.


  Supposing that $\mathsf{deg}_{1/3}(f)\geq(\mathsf{deg}_0(f))^b$ with some $b\in(0.00001,1]$, are there constants $a_f\in(0,0.00001]$ such that,
  $\mathsf{deg}_\epsilon(f) \geq(\mathsf{deg}_0(f))^{a_f}$ at every $\epsilon\in(0,1)$?
  
  Or is there an $f$ such that 
  $\apxdeg_\epsilon(f) \leq \mathsf{polylog}(\apxdeg_0(f))$ at some $\epsilon\in(0,1)$?
  
  Note that $a_f$ is unlikely to be independent of $f$. A closely related question corresponds to what is the highest error $\eta_f&lt;1/2$ as it approaches $1/2$ beyond which no further decrease happens in degree except at exactly $\epsilon=\frac{1}2$.
  
  Nisan-Szegedy result implies only a trivial lower bound of
  $$\lim_{\epsilon\rightarrow\frac{1}2^-}\mathsf{deg}_\epsilon(f)\geq 1.$$


For symmetric $f$ we can have $a_f$ close to $0.5$.

Is there a class of functions where a non-trivial lower-bound is not known?

Is there a class of functions it is conjectured approximate degree $\mathsf{deg}_{ \epsilon}(f)$ at some $\epsilon\in(0,\frac{1}2)$ can be polylogarithmically smaller than the exact degree given that $\mathsf{deg}_{1/3}(f)\geq(\mathsf{deg}_0(f))^b$ with some $b\in(0.00001,1]$?





When error approaches $1/2$. The bound is $\deg_{0}(f)\leq C_\epsilon(\deg_{\epsilon}(f))^8$ where $C_\epsilon$ depends on $\epsilon$. As $\epsilon$ approaches $0$, $C_\epsilon\rightarrow\infty$. So it is possible the upper bound is because  $C_\epsilon\rightarrow\infty$ and $\deg_{\epsilon}(f)\rightarrow1$. This corresponds to $a_f=0$ in the question. Can this degenerate situation actually happen in a boolean function $f$?
If I understand correctly you are asking about the relationship between the degree necessary for exact representation and the degree necessary for approximate representation.  The seminal paper by Nisan and Szegedy shows that for $\epsilon$ constant these degrees are polynomially related.