Good question! In fact, this problem is much easier than 6.7 from the examples because of a minor (but important) difference in the sequence definitions.

In 6.7, the sequence is $a_1 = 1$ and $a_{n+1} = 3^{a_n}$. This means the first few terms of the sequence are$$1, 3, 9, 27, 81, \ldots$$The problem then asks us to calculate $a_{10} \pmod{7}$.

However, in 6.27, we take the mod as part of the sequence, meaning the first few terms are $2, 5, 1, \ldots$. As a hint, think about how this helps us look for a pattern.