I played a bunch of HoI 2 back in the day with a friend of mine, and we were just talking about playing some more and picking up HoI 3, so this sale is perfectly timed. Is there any 'must-have' dlc in addition to the base game, or is the base game for $2,50 sufficient?
Get the three main expansions: Sempire Fi, For the Motherland, and Their Finest Hour.
Reason is that most people on the forums here will assume that you use their finest hour. Whenever people talk about the game on the forums, they will take it for granted that you have those dlcs. For a period of time, I played just with semper fi in addition to the base game. It is fairly different to their finest hour in the terms of tuning of the AI in regards to different countries. Another basic thing that is drastically different is unit composition. The way that hardness/ softness of a unit calculated is a bit different that I remember from the earlier versions of the game when it comes to calculating a combined arms bonus. Also, if you ever plan on playing a sort of multiplayer with others, their finest hour (with the previous two dlcs) is what people base their games off of. There might be a way for you to join those games without the dlcs (I'm not sure because everyone I have come across them) but its best to have them and know what in them (if you want to play multiplayer with other random people). While I'm sure that if you look hard enough you might be able to find someone that only plays the base game, from my experience, most people play with those three main dlcs.
As for the dlcs like sprite packs, I have personally only seen one person play with the sprites instead of the unit counters. It is such a problem for paradox that they are forcing customers in HOI 4 to use the models for units so they have a better shot at selling dlcs based on unit models. Have they confirmed this is the reason? No, but anyone with basic business sense can pick up on it immediately and see the reason.
I hope this aids you in your decision. If you have any more questions feel free to shoot and I'll give you my take on it.