Applecare

Sorry, no. The current rule for AppleCare in the U.S. is that it can only be purchased within 60 days of buying a new Mac. See: AppleCare Products - Mac - Apple

Extended warranty:

Some manufacturers offer extended warranties, also known as service contracts. For a modest pre-payment, the manufacturer agrees to service their product for an additional period of time -- to repair or replace defects in Materials or Workmanship for an additional period. In some cases, the service contract covers additional items beyond defects in materials or workmanship. In some cases, there is a copayment required to obtain this service.

Some users will have No claims during this period. This "insurance effect" allows the manufacturer to keep the price of this service contract relatively low. You generally can not sign up later, because then only owners with problems would sign up, after they discovered an issue.