What you want is a logical sequence of correct changes (or, as correct as anyone could tell at the time; of course no one's perfect).
If you you have to do code review, track down a bug by bisecting a commit history, or figure out what patches from one branch need to be ported to another, you want to have good history. False starts and fixes to typos from previous patches have no value; in fact, they have negative value, as they obscure the interesting information that a good history provides.
Cleaning up history really doesn't take that long. When something is about ready to merge, take a quick look through the history to figure out which patches are redundant or logically belong as part of previous patches, do a "git rebase -i", and squash them into the appropriate patches. In the process, make sure your commit message are actually good enough that someone doing a code review can actually follow what you're doing (no "fixed a bug in this function; fixed a bug in that function"; actually explain what you fixed and why your fix is the right one).
> make sure your commit message are actually good enough that someone doing a code review can actually follow what you're doing
Yes, this is a very important point for rebasing.