I hear it daily. "Always protected by a UV filter" in the for sale adds. "UV filter saved my lens." Etc. It seems people are terrified they might actually take a photograph on a lens that doesn't have a UV filter on it. And absolutely for certain when you buy a lens the salesman (online or in store) makes you feel like you just crawled out from under a rock if you don't buy a UV filter to protect your investment.
Lets get the last part out of the way first: if you aren't aware, UV filters are among the highest profit margin items in most camera stores. The store often makes more selling the filter than they did selling the lens (lenses are a cutthroat business with very little markup). I buy filters in bulk and pay less than half of camera store prices so I know the profit margin is at least 50% for them, more at the huge stores. Lens markup is often 5% at most. So EVERY camera store and salesman is going to try to sell you the filter. That's where the money is.
Lets get the second part out of the way: UV filters prevent fingerprints, wet spray, and blowing sand and grit from impacting the front element. They protect THE COATING ON THE FRONT ELEMENT. That's all (except possibly a few Canon lenses that are reported to have a complete seal only with a filter in place like the 17-55 f2.8 IS). They don't protect the glass from shattering (actually, filters which are soft glass are more likely to shatter and scratch the front element). They don't protect the lens from breaking if its dropped (a lens hood may if its not a wide angle lens, a filter doesn't). And if a lens drops 5 feet and doesn't appear damaged - well I'd check my calibration pretty carefully if I were you.
And here's where the myth really gets busted: You're not using a filter to protect a $1,800 lens. You're using it to protect a $300 front element. It changes the math considerably. Plus you can't use that cheap $25 UV filter - that's going to get a major drop in image quality. Why pay $1200 for a lens and then ruin the image quality with a cheap filter? Very good multicoated filters have little (not none, little) effect on image quality, but a good 77mm filter will run $70-90 dollars). So now we're spending $80 to protect a $300 front element. And filters, being soft glass, not the tempered glass of a front element aren't going to last as long as a lens, so they'll end up needing replacement every couple of years.
Here's the quick math from Lensrentals.com. Over 700 lenses were rented for a total of 16,000 weeks of use (heavy use) resulted in 6 damaged front elements in the last year. Cost to replace front elements was $2,255. Cost of 700 UV filters at wholesale $28,000 for a decent, not great, filter (your cost would be higher per filter, we can buy by the gross). So if we spent $28,000 for filters we would have saved $2,255 in repairs (and this assumes it would have prevented all 6 damage incidents which is unlikely). Does this apply to the individual? Sure. If you buy filters for 5 nice lenses it should cost you at least $275. And it might save you a $350 front element repair someday. Would you pay $275 an insurance policy that covered a maximum benefit of $350? Me neither.