Please be advised as is pointed out below, the video was incorrect and this: $$ x=e^{\frac{\pi}{2}} \Rightarrow x^{x^{x^{x^{...}}}} = i$$Is completely false!
I recently watched the video by real^real^real^... = imaginary? by blackpenredpen and he shows that this is possible:
$$ x=e^{\frac{\pi}{2}} \Rightarrow x^{x^{x^{x^{...}}}} = i$$
This made me wonder if it is possible to find a similar real number for repeated multiplication rather than exponentiation? $$x\cdot x\cdot x\cdot x\cdot x\ ...\ =\ i, x \in \mathbb{R}$$
My initial thoughts were that repeated multiplication is just exponentiation so maybe we could look at the problem like this:
$$\lim_{n\rightarrow \infty}x^n = i, x \in \mathbb{R} $$
So is this possible? If not it would be nice to see a proof.