Now I Know My ABCs

Calculus Level 2

Given that $$\color{green}{a}$$, $$\color{red}{b}$$ and $$\color{blue}{c}$$ are nonnegative real numbers such that $\Large \color{green}{a}^{\color{red}{b}^{\color{blue}{c}}}=\color{red}{b}^{\color{blue}{c}},$ find the maximum possible value of $$\color{green}{a}$$.

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