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Ten.
10 to the 60th power
A googol is 10 to the power of 100, and a googolplex is 10 to the power of a googol (or 10 to the power of 10^100). A googolplexian is 10 to the power of a googolplex, which is a number so large it's practically incomprehensible. The number of zeros in a googolplexian would be a 1 followed by a googolplex number of zeros.
A googol is 10 to the power of 100, which is a 1 followed by 100 zeros. A googolplex is 10 to the power of a googol, or 10^10^100. A googolplexplex is 10 to the power of a googolplex, or 10^10^(10^100). Adding more "plexes" exponentially increases the number of zeros, so a googolplexplexplexplex would have an incomprehensibly large number of zeros.
One hundred zeroes.