What oils are "required" for a decent home kitchen?
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).