Not anymore. You buy Vanilla and you get BC automatically. not that you'd really be playing vanilla even if that is the only one you had, you would just be playing the current expansion, but level capped at 60 and unable to go to Outland, Northrend, the Cata zones, or Pandaria.
As of sept. 19th you also get wrath of the lich king as well.
Haven't yet ventured to the frozen reaches of Northrend? Beginning today, September 19, players are now able to get the original World of Warcraft, The Burning Crusade, and Wrath of the Lich King for only $19.99 as part of the newly upgraded digital Battle Chest, available now on Battle.net. In addition, all World of Warcraft subscribers will now automatically have access to all of the content and features of both The Burning Crusade and Wrath of the Lich King, even if they never purchased either expansion, at no additional cost. It's time to sail north, adventurer - the Lich King awaits!
You also won't be able to get any of the xpac's advantages like the new professions and classes, however since Cata you'll be living in a post Cata world.... kinda odd i know but heck.
Personally its best to buy the Battlechest, get to 60 buy the next, get to 70, buy the next, get to 85. It'll make it worth your time and money to do this so you don't miss anything if you don't want. Otherwise it'll be 20 (Vanilla, BC, WotLK) then 60 (Cata, but check on Amazon for less), and then MoP (it'll go down after release from people just selling xpac codes). You'll roughly get 120 days free with all these xpacs, or you can sign up for a free up to lvl 20 account to see if you like it or not. See Battle.net for more information about that.