If a player controls Doubling Season when a planeswalker enters the battlefield, does it enter with double its starting loyalty?

Doubling Season reads: 'If an effect would place one or more counters on a permanent you control, it places twice that many of those counters on that permanent instead.' This is a replacement effect (CR 614.1) that modifies how counters are placed.

When a planeswalker enters the battlefield, its printed loyalty number defines how many loyalty counters are placed on it as part of it entering (CR 306.5b). This placement of counters is an effect that Doubling Season intercepts and doubles, so the planeswalker enters with twice its printed loyalty.

This interaction is significant because planeswalkers typically enter with enough loyalty to immediately activate their ultimate abilities under Doubling Season — for example, a planeswalker with a starting loyalty of 4 and an ultimate that costs −8 enters with 8 loyalty, making the ultimate immediately usable.

Concrete example: You control Doubling Season and cast Jace, the Mind Sculptor (printed loyalty 3). Instead of 3 loyalty counters, the replacement effect applies and Jace enters with 6 loyalty counters. You can immediately activate his −12 ability if the cost is met (though his actual ultimate is −12, so 6 wouldn't reach it — a planeswalker like Elspeth, Knight-Errant with printed loyalty 4 enters with 8 counters, immediately enabling her −8 ultimate).

Note that this only applies to planeswalkers you control. If an opponent controls Doubling Season, it does not affect your planeswalkers entering the battlefield (CR 614.1).