If you control two Doubling Season effects, does a planeswalker enter with four times its loyalty?

When a planeswalker enters the battlefield, its loyalty counters are placed on it as part of that process (CR 306.5b). Doubling Season's replacement effect states that if an effect would place one or more counters on a permanent you control, it places twice that many instead (CR 614.1a — replacement effects modify how events happen).

With two Doubling Season effects in play, both apply as replacement effects to the same event. Under CR 616.1, if two or more replacement effects would modify the same event, the affected player (or controller of the affected permanent) chooses the order in which to apply them. Each Doubling Season doubles the counters independently and sequentially.

Applying them one after the other: the printed loyalty (say, 4) is doubled to 8 by the first Doubling Season, then doubled again to 16 by the second. The result is 4× the printed loyalty, regardless of which order you choose to apply them, since multiplication is commutative.

Concrete example: You control two Doubling Seasons and play Jace, the Mind Sculptor (printed loyalty 3). Normally he'd enter with 3 loyalty. The first Doubling Season doubles that to 6; the second doubles 6 to 12. Jace enters with 12 loyalty counters.

This is not a special rule about planeswalkers — the same math applies to any counter-placement effect while you control two Doubling Seasons. The key rules are CR 306.5b (planeswalker loyalty placement), CR 614.1a (replacement effects), and CR 616.1 (ordering multiple replacement effects).