Answer
For a simple pendulum, increasing the amplitude increases the distance that the pendulum must travel, however, because the pendulum ends up moving higher, it also increases the total and average acceleration that it undergoes while it oscillates. These two phenomenon actually end up canceling each other out, and the period remains unchanged.
The same is true as well for the physical pendulum, as evidenced by the fact that the angular frequency does not depend on the amplitude.
Work Step by Step
First take the definition of the period of a simple pendulum with amplitude A:
$T=2 \pi \sqrt {\frac{L}{g}}(1+\frac{1}{2^2}sin^2(\frac{A}{2})+\frac{3^2}{2^2 4^2}sin^4(\frac{A}{2}).....)$
By definition, A is always small for a simple pendulum, therefore $sin(A/2)\approx0$ for all A's, regardless of there size. This reduces the formula to:
$T=2 \pi \sqrt {\frac{L}{g}}(1+0+0+0+...)=2 \pi \sqrt {\frac{L}{g}}$
Therefore, the perod is independent of the amplitude
