Is there an equation that will graph the shape or predict the maximum amplitude of a single sample impulse fed through a One-Pole LPF?
As mentioned in the comments, an impulse response with the shape shown in your question can only be obtained by a system with two real-valued poles. In continuous time, with two distinct poles, the total impulse response is
$$h(t)=\frac{e^{-\alpha_1t}-e^{-\alpha_2t}}{\alpha_2-\alpha_1}u(t)\tag{1}$$
It's a basic exercise to determine the location of the maximum of $(1)$, and the result is
$$t_{max}=\frac{\log\left(\frac{\alpha_1}{\alpha_2}\right)}{\alpha_1-\alpha_2},\qquad\alpha_1\neq\alpha_2\tag{2}$$
For $\alpha_1=\alpha_2=\alpha$, the total impulse response is
$$h(t)=te^{-\alpha t}u(t)\tag{3}$$
and the location of the maximum is
$$t_{max}=\frac{1}{\alpha}\tag{4}$$
The figure below shows an example of an impulse response $(1)$ with $\alpha_1=0.4$ and $\alpha_2=0.6$, and with the location of the maximum predicted by $(2)$.