The answer is a rather shocking 190 tabs. How do I know this? Press and hold the 'tabs' button in mobile Safari and it will give you a count:
Hat tip to @swyx (Shawn Swyx Wang) on Twitter for telling the world how to find this out - I've been an iPhone user for years and it's a new one on me.
And going by Shawn's recent Twitter poll, I'm in something of a minority. In fact, most users have less than 10 tabs open:
Number of open tabs in your phone browser right now:
(mobile Safari users: press and hold on the tabs button and it’ll give you a count)
— shawn swyx wang 🗽 (@swyx) November 24, 2018
This whole thing reminds me of a rather terrible joke (which I made up):
"Doctor, doctor, I've got too many tabs open." "I'm afraid you've got taberculosis."