First daughter applied to 8. This was the college app round where I stayed hands-off, thinking it would all work out just fine if she dealt with it independently. It did turn out "just fine", but we could definitely done better financially had I been educated on the modern college realities and paid attention. Even with the GI Bill, this was not easy financially.
Second daughter applied to 19. An extreme number, but there were reasonable factors at play. We live geographically distant from most schools and free time is at a premium (not to mention travel money), so pre-app visits were not logistically realistic, and it did not make sense to apply to schools where she might not be accepted. She was very uncertain about the type of school she wanted as far as size and location, so narrowing it down by those factors was not helpful.
She had the stats to put her in the running for major competitive merit scholarships, but those are never guaranteed. The few schools where we would qualify for sufficient aid had very low acceptance rates as well. Many of her apps fell into those categories.
In the end, she chose her EA school, a suburban East Coast private university. But it was a tough decision. She came extremely close (deciding just before the deadline) to choosing the huge urban Southern state school that offered her a full ride, and there was a small private rural Christian school that stayed in the running until close to the end.
She was selected for some scholarships, and turned down flat for others (that oddly, you would have thought were less competitive). Applying widely was probably the best thing for her situation.
I doubt that will be the approach we take with my youngest child though. Her stats are likely to be different, and she will probably have a better idea about what she wants.