More Fun With Power Towers

Calculus Level 2

aaa=b\huge a^{a^{a^{\cdot^{\cdot^\cdot}}}}=b

Find the largest real number bb such that the above equation has a positive real solution aa. Approximate bb to four significant digits.

If you come to the conclusion that no such value of bb exists, enter 666.

Clarification: The value of the infinite power tower aaa\large a^{a^{a^{\cdot^{\cdot^\cdot}}}} is defined as the limit of the sequence x0=a,xn+1=axnx_0=a, x_{n+1}=a^{x_n}.

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