I wouldn't be too enthusiastic about the merits of playing just the base game, unless as a sort of demo for a limited time, even though it's certainly doable. This is because you are better off not spending time learning mechanics that will be changed the moment you do actually buy those DLCs, which chances are you will.
Still, you can safely skip all of the following until you decide you want to play as whatever type of ruler they unlock:
- cosmetics: songs, unit models, portraits, dynastic shields etc.
- utilities: Ruler Customizer, EU4 Converter etc.
- Sunset Invasion, if you don't want to see Aztecs invading Europe (the only thing this does)
- Sword of Islam, if you don't want to play as a Muslim character (the only thing the DLC unlocks or changes)
- The Republic, if you don't want to play as a Republican character (which is not your ideal starter choice anyway)
- Rajas of India, if you don't want to play in India (the only thing this does, except for the ability to change your religion and culture to your capital city's)
This all should save you some big buck.
On the other hand you
absolutely need the following DLCs:
- Legacy of Rome (only for Retinues, as it makes no point learning to play without them, due to their relevance and the way they affect your ability to withstand the Mongols; otherwise this would be a perfectly optional Byzantine flavour enhancement)
- Sons of Abraham (mechanics relating to Abrahamic religions, a lot of them; it makes no point developing a play style that fails to take advantage of them, and the DLC isn't expensive)
You
almost absolutely need the following DLC:
- Horse Lords (nomadic government is just too fun to live without, even if you don't play it, which you shouldn't initially; and give up Silk Road? c'mon, no way!)
Whether you need the following DLCs depends on you alone (you can live without them, though the options are nice to have):
- Charlemagne (custom kingdoms and empires, viceroys — note that you can do without these; the 769 starting date, which is the earliest available)
- The Old Gods (you may not want to play as a pagan ruler, but the ability to convert pagans to Christianity or Islam is pretty significant; the 867 start date is cool to have; I don't remember to what extent this one affects Vikings along with their raiding/river-sailing special abilities)
- Conclave (council mechanics, changed child education, IMHO detrimental changes compared to base game, but I try to learn to play them to keep in touch with the mainstream game rather than living in denial)
- Way of Life (focus mechanic, which isn't necessarily that much better than old character ambitions, and it adds the whole Seduction mess, which is one of Paradox's most shameful screwups ever, especially considering they haven't fixed it for 1.5 years now)
Please note that at this stage it's difficult to remember/check what exactly is enabled by the paid DLC and what is patch matter, given how DLCs have been accompanied by patches and there were later patches etc.
Wikipedia article you can use as a cheat sheet, though it's not exhaustive.
In other words:
1. It's a totally bad idea to not have Legacy of Rome.
2. It's an extremely bad idea to not have Sons of Abraham.
3. It's a poor idea (IMHO, subjectively) to not have Horse Lords.
4. It's better to have Charlemagne than not have it.
5. It's probably better to have The Old Gods than not have it.
(all of the above will be included in any sort of complete edition)
6. The whole rest is optional. Depending on personal tastes some of them will be must-haves to some people. For example I can't image skipping historically accurate 'Dynastic Shields' (family coats of arms), and I'd rather not be unable to create a custom kingdom or empire. But some of them I use just because I have them, not because I need or want them.
Songs are in the unique position of being extremely worth it (at least the 99% made by that genius dude Andreas Waldetoft) but not gimping your game if you don't have them, as the default set is rich and pleasant enough anyway. It's a bit similar for unit models and even portraits, but I don't want to elaborate further on this because it's subjective. In short, you should judge on your own after carefully inspecting the screenshots on Steam.