A subexponential bound is available (using only things known 40 years ago).
Thanks to John Palmieri over here for pointing out that the $E_1$ term of the May spectral sequence is a commutative polynomial algebra and so ought to have graded dimension which counts some sort of partition, and for subsequently pointing out that on account of $h_{1,0}$, this observation must be supplemented with some information about vanishing lines in the Adams spectral sequence.
Indeed, the May $E_1$ term $V^{\ast\ast\ast}$ is a polynomial algebra in $h_{ij}$ for $i\geq 1, j \geq 0$, with tridegree $|h_{ij}| =(s,t,u) = (1,2^j(2^i-1),i)$, i.e. bidegree $(s,t) = (1,2^j(2^i-1))$ in the Adams $E_2$, i.e. degree $t-s = 2^j(2^i-1)-1$ in the stable stems. Note that $h_{1,0}$ has bidegree $(s,t) = (1,1)$; all other $h_{i,j}$'s have $t-s > 0$.
Let $W^{\ast\ast\ast} \subseteq V^{\ast\ast\ast}$ be the subalgebra generated by the $h_{i,j}$'s other than $h_{1,0}$. Keeping just the last grading $k = t-s$, we see that $\dim W^k$ counts the number of ways of partitioning $k$ using positive integers of the form $2^j(2^i-1)-1$, i.e. positive integers whose binary expression contains exactly one zero (since the numbers $i$ and $j$ are uniquely determined by the quantity $2^j(2^i-1)-1$). This is less than the total number of partitions, and hence subexponential. The estimate via total partitions tells us that $\dim W^k \leq \exp(c\sqrt{k})$ for some $c>0$, but since the allowed parts for partitioning are exponentially sparse like in the Steenrod algebra, I'd guess that an upper bound of the form $\dim W^k \leq \exp(c(\log k)^2)$ actually follows from this if one works through the combinatorics, just as it does with the Steenrod algebra.
Adding back in the generator $h_{1,0}$, we see that $V^k$ is infinite-dimensional for all $k$. But because the $s$-grading of $h_{1,0}$ is still positive, we can use the fact that the $E_2$ page of the Adams spectral sequence (which I'm just calling $Ext^{\ast,\ast}$) has a vanishing line, in the sense that $Ext^{s,t} = 0$ for $0 < t-s < 2s + d$ for some constant $d$. I believe this implies that every element of $Ext^{s,t}$ with $t-s = k$ can be written as a sum of monomials $h_{1,0}^a\prod h_{i,j}^{b_{i,j}}$ where $a < k/2 - d/2$, so that we essentially have $\dim (\oplus_{t-s = k} Ext^{s,t}) \leq \sum_{a=0}^{k/2} \exp(c\sqrt{k-a})$. Because subexponential functions are closed under integration, it follows that the Adams $E_2$ page already has subexponential growth, and hence that $\log_2 |\pi_k\mathbb S_{(2)}|$ likewise has subexponential growth.
There is an odd primary analog of this too. See Ravenel's green book for a version of the May spectral sequence at odd primes where the $E_1$ term is a commutative polynomial algebra.
The dimensions of the graded parts of $W^k$ start off, if I coded things correctly, as:
degree $k$ |
bound on $\dim W^k$ |
0 |
1 |
1 |
1 |
2 |
2 |
3 |
3 |
4 |
5 |
5 |
7 |
6 |
11 |
7 |
15 |
8 |
21 |
9 |
28 |
10 |
38 |
11 |
49 |
12 |
65 |
13 |
83 |
14 |
107 |
15 |
136 |
16 |
172 |
17 |
215 |
18 |
269 |
19 |
332 |
This sequence must be well-known, as I think it is the basis for Bruner's Ext software. So I'm surprised that I can't find it in OEIS. Perhaps I have made a mistake.
Here is an earlier version of this answer:
Thanks to Nicholas Kuhn over here for pointing out that the dimension of the Lambda algebra can be used to bound the size of the $E_2$ term of the Adams spectral sequence.
The dimensions of the graded pieces of the Lambda algebra appear in the OEIS: https://oeis.org/A049285 and a bit of searching reveals that Tangora computed the asymptotics in Level number sequences of trees and the Lambda algebra, where I think he also considers the odd-primary case. Unfortunately, the results are stated in terms of generating functions, and a quick look has not allowed me to find a place where he actually states the asymptotic upper bound he gets on the dimension of the Lambda algebra. But his work is based on Flajolet and Prodinger's Level number sequences for trees, and if I'm deciphering things correctly, it looks like the bound is still exponential.