what type of deep sea fishing is the best/most fun?

no real idea, but I gonna make a guess and say barracuda, out in the sea of course, and waaaaay too much.
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The most fun, would be wherever the fish are biting…they can all be fun from catching Mackeral, to catching Sailfish. Depending on the boat your on, the group of guys your with, etc….And the price…from 100.oo to 1,000.oo.
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"Best" and "most fun" are judgment calls, and as I don't know what you like, I can't answer for you. Some people like trolling big heavy gear dozens of miles offshore for huge marlin or tuna, some like dragging baits along the bottom for halibut, still fishing for bottom fish, or casting artificials near the surface for any of a huge number of species. Some people enjoy almost any kind of fishing you can imagine. I even took a trip a few years ago where we were fly fishing for tuna.

Then you have to decide what is "best" for you: lots of fish, big fish, good-eating or good-fighting fish.

Costs will vary depending on the trip, season, and location. In southern California, you can get on an "open party" half-day boat for under $50, while an offshore trip (open or charter) can run several hundred dollars per person for a day (and there are "long range" trips which can go out for up to a couple weeks or more, which can run several hundred dollars a day.) You'll generally get "better" fishing on the longer trips, but there is plenty of good fishing for smaller fish on the shorter trips, always with the chance for a standout catch.

I would assume the situation is similar in other parts of the world, though the prices may vary considerably.

But be warned — what they like to show on TV is generally the best fishing they can get footage of. Don't expect it to be that good all the time.
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if you live next to the left coast there are many places to get some excellent fishing done almost every seaport has half day, 3/4 day and even twenty four hour turn arounds for tuna (especially San Diego) half day is about $40.00. 3/4 around $60 and over night $100 the water is still cold so I would wait till mid summer to mid fall for the long rides but the half days go out and target bass, sheephead, perch, barries and halibut. and that's in gear right now.
Remember that most fishing shows film from three, even four hours to a couple of days to get enough film for a half hour show. But fishing on the boats is a numbers game you do it and every time you want to learn three secrets that other anglers will share with you. Just remember to write them all down and you will be the teacher after ten trips.
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I love fishing halibut it is pure joy watching a door come out of the mirky water, but white seabass are at the islands on the moons right now as well….. Watch the numbers growing during the moons (both full and new) and the upwellings along the coast mean the squid are spawning and that too will bring them in close to shore.