Wolfram Alpha gave me the answer to this, but unfortunately Wolfram Alpha doesn't show its work, I can't find a proof anywhere else, and my feeble attempts to show it myself went nowhere. How can it be proven that:
$$L_t\left[I_0\left(2\sqrt t\right)\right] = \frac{e^\tfrac{1}{s}}{s}$$
where $L_t\left[f\right]$ is the (one-sided) Laplace transform of $f$ w.r.t. $t$ and $I_n\left(z\right)$ is the modified Bessel function of the first kind.
$$\begin{align}\int_0^{\infty} dt \, I_0(2 \sqrt{t}) e^{-s t} &= \sum_{k=0}^{\infty} \frac1{(n!)^2}\int_0^{\infty} dt \, t^n \, e^{-s t} \\ &= \sum_{k=0}^{\infty} \frac1{n!} \frac1{s^{n+1}} \\ &= \frac1{s} e^{1/s} \end{align}$$