What is $\lim_{n \to \infty} n a_n$?

To compute $na_n$ we start with $y=17$ and apply $n$ times the transformation $$y_{new}=nL(y/n)=y-y^2/(2n)$$ i.e. $$(y_{new}-y)/\epsilon=-y^2/2$$ where $\epsilon=1/n$. For $n\to \infty$ this means that we're solving the ODE $$y'=-y^2/2$$ with the initial condition $y(0)=17$, and want to know $y(1)$. The solutions of the ODE are $y(x)=2/(x+c)$, $y(0)=17$ gives $c=2/17$, hence $y(1)=34/19$.

edit: In more detail: for a given $n$ we define a function $$y_n:\{0,\frac1n,\frac2n,\frac3n,\dots,1\}\to \mathbb R$$ via $$y_n(0)=17,\quad y\bigl(x+\frac1n\bigr)=y(x)-\frac{y(x)^2}{2n}$$ and then extend $y_n$ to a piecewise-linear function $[0,1]\to \mathbb R$. We have $y_n(1)=na_n$. This function $y_n$ is also the outcome of the Euler approximate method of solving the ODE $y'=-y^2/2$ with the initial condition $y(0)=17$ and with the step $1/n$. The sequence $y_n$ converges to a solution of the ODE - that's true for any ODE whenever the RHS is (say) locally Lipschitz in $y$. Not sure about a textbook reference, here is what I found on MO: https://mathoverflow.net/a/200412/9390