I like to use the tire sizing info available from Tire Rack to see what might fit. It's impossible to say which specific sizes are recommended as they vary by brand and even by tire model within that brand. Solution for me anyway is to decide on the tire brand/model, then pull up sizes and specs to see what range of widths the manufacturer recommends for the various available tire sizes. As you look, consider also that the wheelwells may not support a significantly larger diameter tire, especially if you might possibly want to install tire chains sometime in the tire's life on the car. Pay attention also to the amount of room available for extra carcass width with the steering at full-lock positions, again considering possible tire chain room.
I don't consider our Pilot nearly as much a "sport" as a "utility" vehicle. Buying wider tires for potential better "handling" on the slalom course isn't on the radar. Maybe a wider tire for extra flotation while driving in sand? Not a consideration for me. Maybe a slightly taller or shorter tire to tweak the effective final drive ratio? Maybe but still way down on the list. Narrower and taller for winter performance? Good option for dedicated snow tires. Otherwise, bigger in any dimension usually means heavier tires, and some negative impact on ride quality. All that's left to justify "bigger" then is appearance. Want your Pilot to look a part it doesn't play, like aggressive off-road adventure vehicle or trail boss? Have a ball!