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Let's first deal with the exponent, $777^{77}$ . We have $777^{77}\equiv 1 \pmod{4}$ and $777^{77}\equiv 0\pmod{9}$ since 777 is divisible by 3, so $777^{77}\equiv 9 \pmod{36}$ . Since $\lambda(111)=36$ , we have $7^{777^{77}-1}\equiv 7^8$ $\equiv 16 \pmod{111}$ and, multiplying through with 7, we find $7^{777^{77}}\equiv \boxed{112} \pmod{777}$