Answer
$$e^{e^x}+c$$
Work Step by Step
Given
$$ \int e^{x+e^x}dx=\int e^{x} e^{e^x}dx $$
Let $u= e^x \ \ \Rightarrow \ \ du=e^xdx $, then
\begin{align*}
\int e^{x+e^x}dx&=\int e^{x} e^{e^x}dx\\
&= \int e^{u}du\\
&=e^u+c\\
&=e^{e^x}+c
\end{align*}