A lot. I clear my history regularly but a lot a lot a lot. Mostly cause I scan through most new builds to ensure they are in compliance and maybe even try them out; especially when doing videos.
Thousands. My Mechbay on Smurfy contains 300+ builds and that's only the ones I've kept to work with on my own chassis. I have cleared my caches from time to time or done a complete reinstall or used different units (PC, Phone, Tablet) and I've used Smurfy for a long, long time. It probably helps that I visit a lot of builds that people suggest (I've recruited twenty odd people into MWO, not to mention all the forum aid I've given) and point out small improvements.
Damn, I had such moderate hopes for this thread... Not one tangible response! You might get a count if you select them all.
I have four computers that I access Smurfy from (PC, laptop, work PC and phone) but my main PC has 1115 smurfy visits. With my laptop and work PC, I'd add about 100 each and my phone, maybe 40 or so...
I started counting, by going back through my history on Chrome. I got through about the past week and was already over 200, so it was like: "No way am I going back over the last year and half (give or take a month or two)!" So yeah, if I say on average 200-300 per week, and 4 weeks per month, it'd be about 17,500 or so. That's a rough estimate, and I probably did more when I first started playing MWO, but those extras can account for the random times when I took a break or was away. Just way too many, and I keep going. Like, I just checked three different ones in the three minutes before I started writing this!