I especially like having a graph that tells me I need to sleep more regularly [1] and I also like having location history that tells me I'm a very boring person in the real world (and that my phone's GPS is wildly inaccurate).
Location history isn't "your privacy on the Internet", it's "your privacy".
Something I'd like to do (someone else asked for an e.g.) is look back at programming language searches, as they'll nearly always have the lang in the query. I'd be able to see when I started learning e.g. python and had to google everything, when I picked up other languages, how those searches decreased (or increased) as I learned more. Lots of interesting stuff.
I have no problem with collecting your own search history, storing it locally, and analyzing it (though I doubt it would produce any groundbreaking insights). I just think it's safer if you store it yourself.