How to determine how many shells deep I might be?

Solution 1:

echo $SHLVL

From the bash manpage:

SHLVL Incremented by one each time an instance of bash is started.

Solution 2:

One way is to use pstree:

$ pstree -h
[...]
├─sshd─┬─sshd───sshd───bash───bash───bash───bash───bash───pstree
│      └─sshd───sshd───bash───bash───bash───bash
[...]