Answer: 115.

Solution. Let $x$ be the number of non-empty chests.

Consider the process when Koschei just started placing chests inside each other. Initially, he had 11 empty large chests. Each time he placed 8 smaller chests into a single empty chest, the total number of empty chests increased by 7 ( -1 old empty chest and +8 new empty chests).

This results in the equation $11+7x=102$. Then $x=13$. The total number of chests Koschei has is 115.
Problem 6. Koschei the Deathless has 11 large chests. In some of them lie 8 medium chests. And in some of the medium chests lie 8 small chests. The chests contain nothing else. In total, Koschei has 102 empty chests. How many chests does Koschei have in total?