The simplest combo in general is 3 cards: Gilder Bairn, Banishing Knack, and Gemstone Array works for all planeswalkers.
There are 3 card combos around the The Chain Veil (all you need to add is infinite mana and artifact untapping, such Astral Cornucopia with 4 counters and Filigree Sages, or the combo suggested by @Andrew). However, these combos only work for planeswalkers with a "+x" ability that won't kill either you or your opponent in the process of using them repeatedly, so it's incompatible with planeswalkers like Chandra, Bold Pyromancer, Sarkhan the Mad, and Jace Beleren.
Gilder Bairn, Banishing Knack, and a Gemstone Array works for all planeswalkers. Gilder Bairn creates infinite mana with Gemstone Array (provided you have a way to tap the Bairn) and can also pump a planeswalker to have enough counters to use its ultimate the turn it enters the battlefield. Banishing Knack provides both a way to tap the Bairn and a way to bounce and replay the planeswalker, which causes it to be a new object and thus resets the count on how many times it has used a loyalty ability this turn. In order for this to work, you need sufficient starting counters on the Gemstone Array and some cheap permanent you can cast repeatedly to bounce with the Bairn while building up infinite mana on the Gemstone Array, noting that the planeswalker you plan to play can be this permanent. The number of starting counters you need on the array to combo off is 7 + [the cost of your cheapest non-land permanent].
There are fewer-card combos with particular planeswalkers, the clear winner of which is 1 card plus the planeswalker: Doubling Season with Jace, Cunning Castaway.
You can also do this in Commander with 2 cards if Teferi, Temporal Archmage is your commander. All you need is The Chain Veil and Gilded Lotus on the battlefield (as well as 2 lands) and Teferi in the command zone. This works because The Chain Veil activations apply to later instances of Teferi that will be cast, so by repeatedly killing him via his "-1" ability to untap The Chain Veil and your mana sources, and replaying him from the command zone, you are gaining more mana than you are spending on the extra commander tax. As an added bonus, you get to draw your deck while doing this.
Also, credit to @ArcanistLupus for pointing out that an infinite turn combo in a deck with a shuffleback creature in your deck (such as Emrakul, the Aeons Torn) can technically be a replacement for a combo around The Chain Veil (it allows you to use the ultimate on a planeswalker with a non-lethal "+1" ability). There exist two-card combos that do this, such as Beacon of Tomorrows and Planar Portal.