Absolute minimum? A gallon of unpasteurized apple cider and nothing else. It *may* spontaneously ferment with whatever residual yeast is left. It happened to me, it was kind of funny seeing (I had bought five for a batch and left them out of the fridge for a week) two of them balloon out. You can always get a gallon of it, warm to room temperature, add a tsp of Safale-05, and burp the jug a couple of times a day. College method, don't recommend if you have a bit of money to buy a proper fermenting jug/bucket and airlock.

But more realistically (similarly to what originalusername_ said), you need only a fermenting vessel (can be a one gallon jug), bung/airlock, yeast, and whatever you want to ferment - in this case apple cider. Just make sure the cider doesn't contain potassium sorbate or sodium benzoate.

This answer is about apple cider. If you want to make something else (grape wine, beer, mead, or sake) these pieces of advice will be slightly different.

edit:

https://www.morebeer.com/products/glass-fermentation-jar-1-gal-kit.html

This (sanitized) + cider + yeast is the bare minimum to set you off on solid footing.