The condominium of numbers!

Number Theory Level 5

$\huge\color{#624F41}2^{\color{#D61F06}3^{\color{#EC7300}4^{\color{#20A900}5^{\color{teal}6^{\color{magenta}7^{\color{#302B94}8^{\color{#3D99F6}9}}}}}}}$

Find the last three digits of the number above.