I love massaman, green curry, red curry, yellow curry, etc.
Can any of these be made quickly at home with only 4-5 ingredients?
I think @roux is generally right, curries are like mexican moles, they have lots of spices and are fairly complex. But I do think there are some short-cuts.
The most important components in a curry are sweetness, creaminess, heat, citrus, salt, and depth. I don't know if by simple you also mean you want to use common ingredients, or just 'few' ingredients. This is what I would use if I only had common ingredients:
Coconut milk, cayenne pepper, onion/garlic/ginger, lemon/lime juice, salt/pepper, sugar/honey. I'd also add basil/cilantro/jalapeno for green, tumeric/cumin for yellow, ketchup/tomato paste/chili powder for red.
If you have them, the traditional ingredients you're trying to replicate are lemongrass, lime-leaf, red or green chiles, and fish sauce (or soy sauce).
Yes, you will need to buy a pre-made curry paste if you want to get anywhere near 5 ingredients. Mae Ploy is a reputable brand.
To directly answer your question: no, you can't. However, you could prepare a curry paste at home and then refrigerate it. In my experience, they can last for a month without issue.
This recipe is good for 2-3 woks worth of curry noodles, you can also add coconut milk on the wok towards the end.
Put everything in a food processor and blitz. There's your curry paste. Extra points for adding a handfull of fresh coriander leaves, or the flesh of a mango to the mix.
It sure is possible to make a Thai curry with 5 ingredients. All of the coconut based curries have the same basic ingredients. Jungle curry contains stock instead of coconut cream and has a few more ingredients such as green peppercorns.
The 5 basic ingredients to an authentic Thai curry are:
FYI: Most Thai people don't make the paste from scratch because it is readily available at the markets here in Thailand.
*birds eye chilies
*tumeric powder (and the leaves if you can find it)
optional: ginger/ginger powder and tamarind paste.
You can blitz all the ingredients into a blender make it into paste if you want a thick, concentrated and intense flavour or just slice the ingredients for lightly flavoured curry.
Malaysian yellow curry Similar ingredients to Thai cooking. Malaysia is a multi-cultural society, the cuisine depicts Malay, Chinese and Indian. This recipe is an 'everyday' Malay dish. Using a ready-made paste is rare. Called 'masak lemak'. Works with meat, fish and veg. However most Malays cook the meat and veg together. Eg. yellow chicken curry with carrots, potato (oh yes, potato is a veg in South-East Asia!), cabbage, young mango etc.
Let's look at a few from-scratch, simple options.
Assuming you have salt, water, sugar anyway, so not counting them.
"some Umami source" = shrimp paste, soy sauce, MSG, whatever you prefer.
A basic but tasty tom kha base can be made with coconut milk, lime leaves, lemongrass, galangal, chile peppers, shallot, lime juice, some umami source - thicker versions can be served like a curry, and you get the base sauce from 8 ingredients. Add a protein and/or mushroom (oyster or shiitake work great), maybe some beansprouts.
The simplest "real" curry base would be gaeng kua - paste of chilies, garlic, galangal, lemongrass, cilantro root, shallots, lime zest. You'll need coconut milk, an umami source, and lime leaves and juice. So, 10 for the base (lime gives you zest and juice. Use an organic lime!). Protein, and optionally bamboo sprouts, pineapple, and veg of choice.
One very simple but tasty coconut milk based curry is south indian Olan and the variations you can make of it - just good coconut milk, chile peppers, fresh or frozen curry leaves (not curry powder, not dried curry leaves or leaf powder!), cooked black eye (or kidney) beans, and some diced squash/gourd/pumpkin (original uses winter melon, but it is also great with eg a mix of hokkaido squash and zucchini.) - and you have a great soup or rice accompaniment depending on how thick you make it.
EDIT: Thinking about it, getting some fresh/frozen curry leaves is definitely worth it if possible. As said, they have nothing to do with the spice called "curry powder" nor do they taste of it - but they tend to feature in a lot of curries built on very simple spice combinations. Olan, some types of Thoran, simple versions of keralan vegetable ishtoo (stew ;) ), some other south indian curries, some basic indonesian ones (they use a herb called Daun Salam, similar or even identical to it), various implementations of the great mangalorean pineapple curry - all using curry leaves and all getting a complex flavour from very few spices.
No. Proper curries have far, far more than five ingredients. It is possible to make something sort of kind of similar in the sense that some shrimp and fish served in a broth is like a bouillabaisse, but really it will not be the same. If you really want to go that route, go to a decent supermarket that focuses on Asian foods and buy something pre-made.