I never run out of fresh garlic. Never. I try not to run out of fresh rosemary, but do often because I use so much of it. I use it mostly on roasted potatoes, that's why I keep it in a dried form also. Fresh is always best but dried will work on those.
Fresh Basil is one I try to always have on hand, but that's one that I grow my own. You should look into that, you can get a plant at publix for $2 and if you don't use it often, it will last a looooong time. It's one of the easiest to grow and maintain. Smells good in the kitchen too.
Not herbs or spices, but I hate running low on Sriracha, smoked Gouda cheese, Fresh Parmesan cheese, Olive Oil (2 kinds), real butter, red peppers, onions, tomatoes and pasta. FWIW
Probably left something out but those came to me without even thinking about it.