I care deeply about privacy, which is why I don't store any user-related information.
I only store non-user-related information, such as available two letter domains. I also store the response of certain requests in cache, such as WhoIs lookups. However, I don't know who made the request, and I have not and will not implement functionality to see what is saved in the cache.
I do use CloudFlare, which will discard access logs within 4 hours. I do not log anything myself, as I feel confident that CloudFlare and good caching can sufficiently protect the service, and I don't really care how many people use the service.