There is work by Boedigheimer and Henn that bounds the size of *unstable* homotopy groups of spheres or rather of the number of $p$-local summands (i.e. the dimension after tensoring with $\mathbb{F}_p$). The bound is again exponential, namely $3^{q-n/2}$ for $\mathrm{dim}_{\mathbb{F}_p}\pi_q(S^n)\otimes \mathbb{F}_p$. There is a slight improvement in later work by Henn, but the bound is still exponential as I understand it.

Looking at the data, the growth of the stable homotopy groups seems to be less than exponential though. According to Isaksen's charts (with possible miscounts by myself) the sequence of the first few $k_n$ is:

1 1 3 0 0 1 4 2 3 1 3 0 0 2 6 2 4 4 4 3 2 2 8 2 2 2 3 1 0 1 8 4 5 5
5 1 2 3 9 7 5 5 3 3 7 4 10

Particularly big ones are $k_{15} = 6$, $k_{23} = 8$ and $k_{47} = 10$. The contribution of the image of $J$ is $5$, $4$ and $5$ respectively in these degrees. While the image of $J$ should dominate in low degrees, elements of higher Adams-Novikov filtration should become more and more dominant. All in all, the data does not really look like an exponentially growing sequence, but who knows with our limited knowledge?

Edit: I incorporated Allen Hatcher's corrections to my sequence of $k_n$.