Teachers shouldn't have ugly handwriting

Number Theory Level pending

My math teacher wrote the number \(\large 7^{6^{\square}} \), the rightmost number was omitted because I couldn't decipher his ugly handwriting.

I couldn't for the life of me figure out what should be the missing number, but I do recall that the teacher (truthfully) told us that this number has 7776 proper divisors.

What is this missing number?

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