Planeswalkers will enter with double the normal number of loyalty counters. However, if you activate an ability whose cost has you put loyalty counters on a planeswalker, the number you put on isn't doubled. This is because those counters are put on as a cost, not as an effect.

Doubling Season affects permanents that enter with counters.

Everything that is specified by the effect creating the original token or tokens will also be true about the additional token or tokens created by Doubling Season's replacement effect. For example, if an effect tells you to create a token "tapped and attacking," the additional tokens will also be tapped and attacking.

Battles will enter with double the normal number of defense counters.

If there are two Doubling Seasons on the battlefield, then the number of tokens or counters is four times the original number. If there are three on the battlefield, then the number of tokens or counters is eight times the original number, and so on.