List of keeper cards and why.
Crystal Visa - if you get the three authorized users, you and the three free AU each get $250, easily offsetting the AF. Priority pass for TWO holders that allows unlimited guests.
Chase Hyatt, IHG, Marriot; Club Carlson - free nights/status/or points
JetBlue Barclays Plus - free checked bags, 50% discount food purchased in flight and bonus points each year, etc; plus a ton of points for using it to buy JetBlue flights
Amex Bus Platinum - doubles (and in a year, increases by 50%) value of MR points, Marriot/SPG and Hilton status, Centurion lounge access, etc.
Citi Prestige - generous $300 travel credit that works on airplane tickets, one of the best travel insurance of all the cards, 4th night free benefit. Used to be even better before this month
Chase Reserve/Preferred - someone above already explained why.
Any of the multitude of Amex cards, preferably the ones with no AF - I know it's a broad category, but the Amex cards generally give you the Amex offers, which can save you a ton of money, so any one is a keeper in my mind.
BofAML Alaska Air - besides the normal free checked bag, etc., this has a great annual companion fare where you get a companion ticket for $99 plus taxes, potentially saving you a ton.
Chase Unlimited (good all around), Chase Ink Preferred (good for buying gift cards, paying for cell phones, etc.) and Chase Freedom (good for quarterly category) - keepers for ways to earn high number of UR that are worth a great amount.