As others have said, there are no 'must have' wonders, as you can do well without them, though they're all nice to have!
Something posters forget to mention, though, is that map size has a lot to do with their value. If you decide to build a wonder on a small map, you're devoting a large amount of your total civilization's resources to build 1 thing. On a huge map, you're devoting much less of your civilization's resources (because you have many more cities, but you're still only devoting 1 city's output to the wonder). So the opportunity cost of every wonder goes down as the size of the map increases.
Conversely, most wonders affect every city in your civilization, or every city on your continent, so the benefit of most wonders is proportionally the same as you increase the size of your map. Thus, with most wonders, you're spending less of your total output for the same benefits when your map size increases!
As an example, let's consider the 'average' civ size the number of cities a civilization needs to build 3 cathedrals (a key threshold for culture victories). Then an 'average' civilization on small maps is 6 cities, and on huge maps it's 12 cities. Let's consider the Spiral Minaret. To do the math with real numbers, we'll say your 'average' city is producing 20 hammers per turn and has a monastery and church. We'll also say you've already built 3 cathedrals.
So, on a small map, your 6 cities are producing 120 hammers per turn. So the 550 hammers to produce the Spiral Minaret represent 4.58 turns of output of your entire civilization. On a large map, your 12 cities are producing 240 hammers per turn, so the Spiral Minaret represents 2.29 turns of output of your entire civilization. It's opportunity cost is cut in half!
But what about the benefits? On a small map, you have 15 religious buildings, so you get 30 gold per turn before multipliers. That's 5 gold per city per turn. On a large map, you have 27 religious buildings, so that's 54 gold per turn before multipliers. That comes out to 4.5 gold per turn. (Note, the only reason they're not exact is because you can have more cathedrals per city on small maps...if you were to take a wonder like the Hanging Gardens, the 2 numbers would be equal).
So on small maps, you're devoting 4.58 turns of production to get 5 gold per turn (before multipliers) per city, while on huge maps, you're devoting 2.29 turns of production to get 4.5 gold per turn (before multipliers) per city. It's a much better deal on bigger maps!
Another way to look at it is this. You're spending 550 total hammers for 30 raw gold per turn on small maps. You're spending the same 550 total hammers for 54 raw gold per turn on huge maps.
This reasoning works with most wonders. I like to play on huge maps, and don't generally launch offensive wars, so I tend to try and build wonders in 1 city, occasionally using a 2nd or 3rd when there are multiple ones I want. On a huge map, I still have plenty of cities to focus on military units and developing my economy. When I play smaller maps (which is rare, because I find I simply don't enjoy it as much) I have to remind myself to only go for only an occasional wonder, because your economy or military production will suffer.
It's worth pointing out that there are a few wonders that increase in power on smaller maps. Any wonder that focuses it's affects on 1 city is going to have a bigger effect on smaller maps because you have fewer cities. Most of these wonders are national wonders, however. World Wonders that only affect one city include the Temple of Artemis and the Great Library. Now, both of those are nice to have on large maps (the Great Library is actually great to have on large maps), but they're even nicer to have on small maps, because the benefits are better! I also think Great People are slightly stronger the smaller map you play on, so I believe the Parthenon gives slightly stronger benefits on smaller maps
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Anyway, the above is just a remark to remember when you evaluate what people say on this thread. Few people mention what map size they play at when they respond to threads like this, despite the fact that its an important fact to consider.
As I said, I generally play on huge maps, and the Wonders that have the best benefits for me are probably Pyramids, Parthenon, Apostolic Palace, Christo Redentor, 3 Gorges Dam, and the Space Elevator.
Pyramids- early representation is really nice, and the Great Engineer GP points are nice. I get this maybe half my games
Parthenon- more Great people are really nice, but I usually don't focus on the tech, so I don't get it too often.
Apostolic Palace-can be mostly a non-factor in the game, but even then it gives you 2-6 hammers per city before multipliers!. Other times it can drastically affect the outcome of the game, affecting wars and trading relationships. Again, I don't generally focus on the tech, so I don't get it too often, but its one of the few early wonders I'll use a Great Engineer if one pops out around the time I get the tech. (I generally settle early GE's to increase my chance of getting more late game wonders, even though I might miss out on some of the big early ones).
Christo Redentor-very very powerful. And the AI seems to prioritize it last among the 3 wonders that come with it's tech, so I think I've gotten it each of the BTS games I've played so far that have gone that long!
3 Gorges Dam- If your cities are focused on one continent, it gives you less polluting energy to a lot of cities. Not something I'd probably recommend on smaller maps
Space Elevator- going for space victories, it gives you a nice boost. On a large map, you can just about build every spaceship part simultaneously in a separate city. So it's worthwhile to rush to the 2 tech requirements for this wonder before going back to get all the other techs needed to build your spaceships parts. Remember, you can't use great engineers to hurry spaceship parts or the internet, but you can use your great engineers to rush this! So once the corporations are founded and the space race starts, this is what your great engineers should be saved for (it takes 2 great engineers, and even then you still have to add more hammers). Once it's built, use other spare great people to start a golden age if you can to rush through your space ship production.
