First, let's factor out the part we know, then simplify $ \begin{align} \lim_{n \rightarrow \infty } \left( 1 + \frac{1}{n} + a_{n}\right)^{n} &= \lim_{n \rightarrow \infty } \left( 1 + \frac{1}{n} \right)^{n} \left( \frac{1 + \frac{1}{n} + a_{n}}{1 + \frac{1}{n}} \right)^{n} \\ &= \lim_{n \rightarrow \infty } \left( 1 + \frac{1}{n} \right)^{n} \lim_{n \rightarrow \infty } \left( \frac{1 + \frac{1}{n} + a_{n}}{1 + \frac{1}{n}} \right)^{n} \\ &=e \lim_{n \rightarrow \infty } \left(1 + \frac{n a_{n}}{n + 1} \right)^{n} \end{align}$
What's left looks sort of like the usual limit for $e$, so let's fiddle with it
$ \begin{align} \cdots &=e \lim_{n \rightarrow \infty } \left(1 + \frac{(n-1) a_{n-1}}{n} \right)^{n-1} \\ &= e \lim_{n \rightarrow \infty } \left(\left(1 + \frac{(n-1) a_{n-1}}{n} \right)^{n} \right)^{\frac{n-1}{n}} \\ &= e \left( \lim_{n \rightarrow \infty } \left(1 + \frac{(n-1) a_{n-1}}{n} \right)^{n} \right)^{\lim_{n \rightarrow \infty } \frac{n-1}{n}} \\ &= e \left( \lim_{n \rightarrow \infty } \left(1 + \frac{(n-1) a_{n-1}}{n} \right)^{n} \right)^{1} \\ &= e \lim_{n \rightarrow \infty } \left(1 + \frac{(n-1) a_{n-1}}{n} \right)^{n} \end{align} $
Better. This looks even more like the limit for $e^x$; but the part that should be $x$ is going to $0$. So let's call it $L$ and bound it:
$ \begin{align} L &\leq e \lim_{n \rightarrow \infty } \left(1 + \frac{x}{n} \right)^{n} \\ &= e^{1+x} \end{align} $
$ \begin{align} L &\geq e \lim_{n \rightarrow \infty } \left(1 - \frac{x}{n} \right)^{n} \\ &= e^{1-x} \end{align} $
for all $x > 0$. That is, if $L$ is the limit, then
$ e^{1-x} \leq L \leq e^{1+x} $
Now we can use the method of exhaustion (i.e. squeeze theorem):
$ e = \lim_{x \to 0^+} e^{1-x} \leq L \leq \lim_{x \to 0^+} e^{1+x} = e$
and therefore $L = e$.