Will the site load slower when using a sub-theme of a sub-theme?
TL;DR Have as many sub-themes as you want, it's not a problem.

Longer answer All types of inheritance, theming or otherwise, will have a performance penalty, and really all logic that you perform will have that effect; However


With proper caching enabled, only the very first request that uses the sub-sub-theme, is actually going to notice a difference. After that you will have aggregated css/js files from all three (or more) involved themes, and all of this becomes a non-issue until you clear the cache, after which you will again need to aggregate the files once.
With a pretty big generalization, the performance cost is roughly going to be linearly proportional to the number of changes you make. So if you don't make a lot of changes, you won't see a big effect.


Generally speaking, I suggest you approach the problem the other way around. Organize your theme/solve your problem in a way that is efficient for you as a human. Then worry about how to make it efficient for a machine only when necessary.

Since your site isn't written in assembly, your entire site is a huge performance problem, but in the vast majority of cases, a cpu performing literally trillions of calculations in a second makes that a non-issue.