Can you take infinitely many square roots of Pauli-X?

Unitary matrices can be raised to any power, including fractional powers, so any root you want you can find. You find the root by eigendecomposing the matrix, modifying the eigenvalues (raising them to the desired power), then putting the matrix back together.

In the case of the Pauli X matrix, the eigenvectors are $|+ranglelangle +|$ and $|-ranglelangle -|$ so you can find roots like this:

$$X^s = frac{1}{2}begin{bmatrix} 1 & 1 1 &1end{bmatrix} + frac{e^{i pi s}}{2}begin{bmatrix} 1 & -1 -1 & 1end{bmatrix}$$

Once that's done the actual challenge is realizing $X^s$ gates using the gate set you have available on your computer. For example, if you're using the Clifford+T gate set then you could approximate the rotation using a series of H and T gates.

Note that, for performing a many-controlled NOT, there are more efficient ancilla-free constructions than the one you linked: https://algassert.com/circuits/2015/06/22/Using-Quantum-Gates-instead-of-Ancilla-Bits.html