What types of beer respond best to aging?
Lighter-hopped beers, porters, and stouts respond best to aging. Generally, hops and wood don't get along well -- the citrus and vanilla flavors clash worse than plaid and polka dots, so it's a real trick that I&G have managed a Toasted Oak IPA that doesn't taste like a rotten Creamsicle. Jester King, on the other hand, barrels mostly sours, farmhouse ales, and wild yeast: beers whose flavors are largely derived from the unique fermentation variables.
Is the aging process complete once it's bottled/kegged?
Not necessarily. When bottled, the wood-aged effects of I&G's beers subside for about a month before reasserting themselves. Since Jester King's beers aren't filtered or pasteurized, they're still alive in the bottle and will slowly evolve over time. And of course the whole process isn't truly complete until the beer is in your stomach.
Dan Gentile is a staff writer on Thrillist's national food and drink team. He used to be pretty loose with the "wood" jokes, but over time his writing has become much more mature and slightly stiffer. But not that much stiffer. Get it? Follow him to more zingers at @Dannosphere.