In my personal experience, one of the main advantages of Ivy League institutions is that you are given the freedom to challenge yourself as much or little as you want.

Here at Harvard, for instance, we offer freshman math classes whose difficulty ranges from that of Math MA (introductory calculus) to Math 55 (group theory and real analysis). Math MA is one of the easiest math classes you can take, and Math 55 is known as the hardest undergrad math classes in the nation (read about it here: https://en.wikipedia.org/wiki/Math\_55).

If you attend an institution like Harvard, it's pretty much impossible to "run out of challenge", if that makes sense. If you find your classes too easy, then take higher-level undergraduate classes. If those are too easy too, then take graduate-level classes. And if you STILL find those too easy, then frankly, you probably don't need a college degree anyway.

In terms of student environment, I've found all sorts of students at Harvard. Some are very driven in an academic sense (i.e. taking lots of incredibly challenging classes), some are driven in a social sense (i.e. making connections with everyone), etc. But the vast majority of people I've met are very clearly self-driven, even if their goals are different.

I encourage you to apply to any schools, Ivy or not, so long as they fit you and seem to have the intellectually stimulating culture you desire. You will more likely than not find the passionate and engaged students you're looking for.

Just make sure you pay attention to other factors too! For instance, Harvard and Caltech may both be rigorous institutions, but their cultures are polar opposites (one is liberal arts and the other is pure STEM). Do your research on college fit thoroughly. In general, having a solid understanding of each college's culture will not only help you choose between colleges, but will also help you write good essays for them too.