At a bare minimum, you'll want at least one "flavoured" and one "unflavoured" oil.
Olive oil is definitely the most common choice for a flavoured oil, and sunflower
/peanut
/canola oil all have pretty neutral flavours (not to mention similar smoke points).
Note that for good flavour you'll want extra virgin olive oil, which also makes it unsuitable for frying/sautéing at high temperatures.
So I would say that you already have the minimum requirement.
That said, even the "unflavoured" oils have subtly different flavour characteristics due to their different fat contents, and you'll want to take this into consideration if you like to cook in ethnic cuisines.
A short list (and keep in mind that this is partially subjective):
Peanut oil:
Asian cuisine, Mexican cuisine (high-temperature frying) Sesame oil: Asian cuisine (stir-fry or deep-fry [refined only])
Olive oil:
Italian cuisine, French cuisine (low-temp) Sunflower or Canola oil: American cuisine, esp.
deep-frying Grape seed oil:
Decent substitute for any of the other oils Walnut/
almond oil:
Great for fried desserts (watch the smoke point though).
cooking.stackexchange.com/users/41/Aaronut
If you deep-fry, peanut should be the 2nd.
The neutral seed oils (canola, sunflower, corn, etc.)
all cook about the same.
Peanut has a fairly high smoke point, so is better for deep-frying.
Olive has a good flavor.
Sesame, almond, walnut and other nut oils are short-lived, and tend to be expensive.
They make great additions to salad dressings or as a finishing flavor in some dishes (sesame is great in a stir fry).
Olive oil has a low smoke point (especially extra virgin), and will loose considerable flavor if heated, so it isn't recommended for frying (it costs more than the seed oils, and its a shame to waste it).
Any high-temp cooking should not be done in olive oil.
Since I don't deep fry often, I have two cooking oils on hand: good olive, and canola.
The olive is used mainly to finish or in lower temp saute/sweat.
The canola is used to fry onions, pancakes, meat, and anything else that takes some higher heat.
I chose canola because it has a neutral flavor and a fairly good omega-3 content.
Plus it was cheap.
I also have some white truffle, walnut, and sesame in the fridge for finishing.
They aren't used terribly often, and a bit goes a long way.
Also: butter.
Although it isn't really an oil, I use lots of butter/ghee(clarified butter) in my cooking.
Good one to make sure you always have on hand.
cooking.stackexchange.com/users/624/Adam Shiemke
Olive extra virgin and sunflower.
Same here.
Make sure the olive oil is extra virgin though, for flavor and use sparingly.
cooking.stackexchange.com/users/641/BaffledCook
We often have much more than two, but I'd say the two you really need are olive oil and some form of vegetable oil (vegetable, cannola, sunflower).
The second type will often have a higher smoke point than olive oil.
I find olive oil has much more flavor than vegetable or cannola oil.
Olive oil is often used in dressings as well.
Once you start to get in to more specific dishes, you may find the number balloons from two though.
Sesame oil for asian food, peanut oil for deep frying, a nice evoo for salads.
I think we have about 6 different types of oil, but we mostly just use olive and cannola.
cooking.stackexchange.com/users/1259/yossarian
If you're looking for a 'must keep on hand' list -- a mild oil, extra virgin olive oil and butter will get you through most anything.
Add shortening if you like baking, and sesame oil if you like to cook asian food, and you'll be prepared for most anything.
For the justification see my response to What oil or fat to use for different purposes?
cooking.stackexchange.com/users/67/Joe
I've actually come to find lately that grape seed oil is extremely versatile.
It has a very light, mild flavor, and when used in cooking you almost can't taste it at all.
When used in its "raw" state, the flavor is a little nutty, and makes a very nice addition to salads/dressings.
I've used it to make a basil infusion that was great.
And, with a very high smoke point, it's safe to use for frying and sauteing.
I now use grapeseed oil in place of olive and vegetable oils in a whole lot of recipes.
It's become my go-to oil lately.
If I had to have only two oils in my kitchen, they would be an olive oil, and grape seed oil.
cooking.stackexchange.com/users/1236/stephennmcdonald
One oil with a high smoke point, plus any with flavors you like.
I keep: peanut or canola oil extra virgin olive oil sesame oil and sometime a small quantity of something else if a recipe called for it.
Other fats on hand include: butter schmaltz (skimmed from a chicken pot left to cool, rewarmed, poured into ice cube trays to freeze and kept in a baggie in the freezer) bacon fat (poured hot from the pan into a jar by the stove; mostly used for seasoning my cast iron, but also for flavor)
cooking.stackexchange.com/users/1670/dmckee --- ex-moderator kitten
Olive oil is a must have in a Kitchen.
The only other type of oil that you want to keep is something with a neutral flavor and high smoke point.
I recommend Canola oil or Vegetable oil.
cooking.stackexchange.com/users/145/Covar
What different varieties of oil are  really  needed in a decent home kitchen?  I ask because I've only two right now (olive & sunflower) that I try to use "correctly" but sometimes use interchangeably. I wonder what I'm missing.
What oils are "required" for a decent home kitchen?
cooking.stackexchange.com
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cooking.stackexchange.com/questions/6606
cooking.stackexchange.com/users/1181/DMA57361
You should have at least one "flavoured" oil (e.g. olive) and one "unflavoured" oil (e.g. vegetable, cannola or sunflower). Peanut oil is also useful for deep frying and grapeseed oil is also very versatile. You should have one flavoured oil like olive oil and a neutral vegetable oil like canola or peanut oil for frying.
You should have one flavoured oil like olive oil and a neutral vegetable oil for frying. Try peanut oil, grape seed oil or canola oil.