Julia, 18 bytes

n->floor(log2(n)n)

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input needs to be a BigInt to avoid floating point errors

Julia, 18 24 bytes

n->ndigits(n^n,base=2)-1

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port of Manish Kundu's answer

previous answer:

n->floor(log2(n)n)

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input needs to be a BigInt to avoid floating point errors.

Technically invalid because you need to augment the precision for very big numbers (with setprecision(2048) for example for 10^100)