Essential Asian Ingredients
Asia is vast. Is there a particular cuisine you'd like to focus on? Japanese, Chinese, Korean, Thai, Vietnamese...? There is some overlap, but not everything. If you want a little of everything, get sesame oil, kimchi, sweet soy, mirin, rice wine vinegar, sambal/sriracha, hoisin sauce, oyster sauce.
re: babette feasts
It seems like Vietnamese cuisine relies a lot more on fresh herbs - purple basil, cilantro, culantro, perilla/shiso, & more - than prepared condiments, with vinegar, sugar, chili, lime, and fish sauce on the side. So fish sauce, soy, various chili pastes, white vinegar for making chili or garlic vinegar, and pho restaurants always have hoisin. Also little dried fish and shrimp. For Cambodian maybe some shrimp paste too. Is there anywhere to browse cookbooks and stealthily make a shopping list?
Given your focus on Vietnamese/Cambodian, and supposing that you don't mean to shop there every few days, I'd suggest you get them to recommend a fish sauce, and then get a bunch of freezable produce (galangal root, lemon grass, kaffir lime leaves). There's a whole world of handy prepared chili pastes (I find Maesri and Mae Ploy brands to be pretty reliable). If your thinking trends more in the China direction as well, add some soy sauces (Kimlan brand is my current one) and things like Oyster sauce (Panda brand). Also, you can get coconut milk cheaper at those Asian stores than at run-of-the-mill American markets (Chao-Koh is a good brand).
For nearer-term eating, make sure to grab the usual bargains in shallots, garlic, ginger, various greens and herbs, and bean sprouts. All much cheaper in Asian markets, and generally fresher, too (with the exception, perhaps, of garlic, which is often close to sprouting at my local Asian markets, for some reason).
This is what my mother must always have in her pantry to cook her vietnamese dishes:
-frozen minced lemon grass
-bird's eye chili
-rice noodles for bun
-rice noodles for pho
-dried shiitake mushrooms
-dried mung beans