To solve the integral of $(e^x)^{e^x}$, let's express it more clearly:

$I = \int (e^x)^{e^x} \, dx = \int e^{x e^x} \, dx$

This integral is quite complicated and does not have an elementary antiderivative. It falls under the category of non-elementary integrals, which means it cannot be expressed in terms of basic functions like polynomials, exponentials, logarithms, trigonometric functions, or their inverses.

However, this type of integral might be evaluated using special functions (like the Lambert W function) or numerical methods, depending on the context or further details.

Would you like to explore a numerical approximation for this integral, or dive into special functions like the Lambert W function that could be involved in solving more complex expressions like this?

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Here are 5 follow-up questions to explore this topic further:

1. Do you want to see the numerical solution for a specific interval?
2. Would you like to learn about the Lambert W function and its applications in integrals?
3. Do you need to compute another integral that involves an exponential function?
4. Would you like to explore integration techniques for exponential and logarithmic functions?
5. Do you want help with understanding non-elementary integrals?

Tip: Many complex integrals, especially those involving iterated exponentials or products of exponentials, often don't have simple closed forms and require advanced techniques for evaluation!