Some people attempt to create dishes that deliver combinations of these tastes. These days, you'll hear folks talk about how they really enjoy the combination of sweet and salty. Examples can be found in chocolate-covered pretzels, many trail mixes and the addition of sea salt to items in bakeries like chocolate cookies.
Sweet and sour is popular as well -- and always has been. This is evidenced by the classic sweet and sour chicken that so many of our parents order in Chinese restaurants. Additionally, many fruits could be considered both sweet and sour. For example, oranges and mangoes.
Among many of the fantastic products you can find on the shelves at Trader Joe's, consider the photo above of Chile Spiced Mango. This snack consists of dried mango (a favorite of mine as is) dusted with a generous amount of chili powder mix. These take sweet and sour one step further by introducing 'spicy' to the mix. While 'spicy' isn't one of the five flavors, it's definitely a great addition. After eating the first piece, the spice sticks with you and leaves you craving another.
A great, healthy snack.
Note that these are relatively spicy and may not be appropriate for everyone. But if you like them, you'll love them.