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Such number $x$ statistifies $i^x=x$. This can be solved with Lambert W function. Doing so, gives: $$x \approx -1.861743075013160441612498412-0.4107999688363923527986399873 i$$ $$x \approx 0.4382829367270321593816295045+0.3605924718713854895879688663 i$$ But we see that they are always positive, so it is the second one.


It seems like you only want to focus on towers of exponents and usage of variables. You might assume that the magnitudes of the numbers you can form are always strictly ordered by the heights of the exponential towers (which is a reasonable assumption, since it would take much larger numbers (hence, probably longer coding length) in a shorter tower to give a ...

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