Not recyclable curbside.
Hard: Hybrid Diapers (Disposable Insert + Cloth Diaper)
There are a number of companies that sell cloth diapers that you can insert a disposable insert into. This method doesn’t totally eliminate diaper waste, but definitely cuts down.
Hard: Cloth diapers
I don’t have personal experience with cloth diapering, but I found this blog post and video series to be extremely informative. Using cloth diapers + cloth inserts will eliminate a huge amount of waste.