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### Solution

\begin{aligned} g(x)&=e^{x} \\\\ g(g(g(x)))&=e^{e^{e^{x}}} \\\\ (g(g(g(x))))^{\prime} &=g^{\prime}(g(g(x))) g^{\prime}(g(x)) g^{\prime}(x)\\\\&=e^{e^{e^{x}}} e^{e^{x}} e^{x} \\\\ \int_{0}^{1} e^{e^{e^{x}}} e^{e^{x}} e^{x} d x &=\int_{0}^{1}(g(g(g(x))))^{\prime} d x \\\\ &=[g(g(g(x)))]_{0}^{1} \\\\ &=\left[e^{e^{e^{x}}}\right]_{0}^{1} \\\\ &=e^{e^{e^{1}}}-e^{e^{e^{0}}}=e^{e^{e}}-e^{e} \approx 3.814 \times 10^{6}\\\\ \int_{0}^{1} e^{e^{e^{x}}} e^{e^{x}} e^{x} d x &\approx 3.814 \times 10^{6} \end{aligned}

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