About a half-dozen, with plans to acquire another 4-5 this year for signup bonuses. My wife and I have been signing up separately to each get the bonus. Sometime next year we're going to have a 2-3 week vacation in Europe, including business class flights and 4-5 star hotels every night, that will have been fully subsidized by CC signup bonuses.
Current cards:
Chase Amazon Visa (oldest card)
Chase Freedom
Chase Sapphire Preferred
Chase United MileagePlus Explorer
AmEx Blue Cash Preferred
Discover IT
Target Red Card
On my radar for this year:
Chase Hyatt (2 free nights at any Hyatt property)
Club Carlson Visa (85k points, last night free on stays of 2 nights or more)
AmEx Starwood Preferred Guest (25k points = approx 2 nights)
Coach tickets to Europe are unchanged @ 60k round trip. The new points rates are live on the United website.