Similar to Maxim:

$$ \begin{align} f(a)&=\sum_{n=1}^\infty \frac{a^n}{n^n}=\sum_{n=1}^\infty e^{n \log(a/n)}\\ &\approx \int_1^\infty e^{t \log(a/t)} dt \\ &= a \int_0^a \frac{1}{u^2} e^{a \log(u) /u} du\\ &= a \int_0^a h(u) e^{a g(u)} du\\ &\approx a \sqrt{\frac{2 \pi}{a |g''(u_0)|} } h(u_0) e^{a g(u_0)} \end{align} $$

where we've used Laplace's approximation (assuming $a\to \infty$) to $h(u) =\frac{1}{u^2}$ and $g(u)=\log(u)/u$, with $u_0=e$ , $g''(u_0)=-1/e^3$ . Then the approximation gives

$$f(a)\approx \sqrt{2 \pi a} \exp( a/e-1/2)$$

or

$$\log f(a)\approx \frac{a}{e} + \frac{1}{2}\log(a) + \frac{1}{2}(\log(2 \pi)-1) $$

Similar to Maxim:

$$ \begin{align} f(a)&=\sum_{n=1}^\infty \frac{a^n}{n^n}=\sum_{n=1}^\infty e^{n \log(a/n)}\\ &\approx \int_1^\infty e^{t \log(a/t)} dt \\ &= a \int_0^a \frac{1}{u^2} e^{a \log(u) /u} du\\ &= a \int_0^a h(u) e^{a g(u)} du\\ &\approx a \sqrt{\frac{2 \pi}{a |g''(u_0)|} } h(u_0) e^{a g(u_0)} \end{align} $$

where we've used Laplace's approximation (assuming $a \gg e$) to $h(u) =\frac{1}{u^2}$ and $g(u)=\log(u)/u$, with $u_0=e$ , $g''(u_0)=-1/e^3$ . Then the approximation gives

$$f(a)\approx \sqrt{2 \pi a} \exp( a/e-1/2)$$

or

$$\log f(a)\approx \frac{a}{e} + \frac{1}{2}\log(a) + \frac{1}{2}(\log(2 \pi)-1) $$

I've not done the strict asyptotical analysis, but it looks as if the error is $o(1)$. Some numerical values

 a      log(f(a))    aprox      abs error 
3.0     1.896554    2.071883    0.175329
5.0     2.984687    3.063055    0.078368
7.5     4.150135    4.185486    0.035350
10.0    5.229637    5.249025    0.019389
20.0    9.268130    9.274393    0.006264
30.0    13.151944   13.155920   0.003976
50.0    20.766590   20.768922   0.002332
75.0    30.167102   30.168641   0.001539
100.0   39.508319   39.509468   0.001149
200.0   76.643415   76.643985   0.000570
300.0   113.634283  113.634662  0.000379
500.0   187.465736  187.465963  0.000227
750.0   279.638405  279.638556  0.000151
1000.0  371.752144  371.752257  0.000113