According to this video, $\varphi$ is the most irrational number, due to its continued fraction form having $1$, the smallest natural number, in the denominators.
Is it not possible to construct a "more irrational" number by using $0$?
For example,
$\iota = 1 + \cfrac{1}{0 + \cfrac{1}{1 + \cfrac{1}{0 + \cfrac{1}{1+\cdots} } } }$
Based on the argument in the video, this would appear to be more irrational.
What am I missing?
No because $$\iota = 1 + \cfrac{1}{0 + \cfrac{1}{1 + \cfrac{1}{0 + \cfrac{1}{1+\cdots} } } } = 1 + \cfrac{1}{ \cfrac{1}{1 + \cfrac{1}{ \cfrac{1}{1+\cdots} } } }$$\ $$=1 + 1+\cfrac{1} {\cfrac{1}{1+\cdots}}=1+1+1+\ldots $$