Batteries don't last forever - three years is about average. Standing idle in cold weather really drains them. Remember, when the AA get the car started again, you need to drive it for around 20 minutes to get the battery fully charged again.
20 mins to get the battery fully charged again! It takes 20 minutes' driving to replace a start, let alone charge it fully from flat. Batteries tend to be around 40-50 amp hours capacity, so would need to be charged at 40-50 amps for around an hour to charge from fully flat, without allowing for the fact that the charging is not 100% efficient; that would boil the battery. You need to get it on a battery charger or drive it for quite a while.