There are a lot of integral representations for $\pi$ as well as infinite series, limits, etc. For other transcendental constants as well (like $\gamma$ or $\zeta(3)$).
However, for every definite integral that is equal to $e$ I can think of, the integrated function contains the exponent in some way.
Can you provide some definite integrals that have $e$ as their value (or some elementary function of $e$ that is not a logarithm), without $e$ appearing in any way under the integral or as one of its limits (and without the limits for $e$, or the infinite series for $e$)?
The example or what I want is the following integral for $\pi$:
$$\int_0^{1} \sqrt{1-x^2} dx=\frac{\pi}{4}$$
If you are asking whether e is a period, the question remains, technically, still open, but its answer is not expected to be affirmative. As to the non-algebraic integrands, we have $$\int_{-\infty}^\infty\frac{\cos(ax)}{1+x^2}~dx~=~\frac\pi{e^{|a|}}$$