Yeah, "big bass" doesn't do much to narrow it down. What kind of mids and highs does the OP want? What kind of music does he plan to play on them, and how big is the room? Does he like a laid-back speaker, or is he looking for something more-up-front and in-your-face-sounding? Personally, I would go with the best vintage TOTL model I can find for under a grand (because TOTL models usually get that extra attention at the design/build stage). If you want big bass in one speaker without a separate sub, you're looking at something BIG (Something big that will work well in your room... Some large, big-bass speakers need a big room to truly-shine while others do better in smaller rooms). So, if you factor in the budget, that limits your options. The well-known brands like Altec and JBL are gonna' be slim for under $1000, so why not go for something below the radar? Model 19's would have the big bass and smooth, balanced, open sound, but I doubt you'll even find them for $1500, let alone a grand (unless the cabs are totally-beat or something... The drivers/crossovers alone would probably fetch more than a grand). Anyway, just a suggestion. Just saying what I'd do in that situation...
And I agree that even 10" woofers can sound big if you have enough of them in one enclosure. 10" woofers can also put out very-SMOOTH bass while still retaining a fast response (The 2x10" enclosure is my favorite for guitar amps for that reason). I like 12" woofers for hi-fi. They seem to be a happy medium between smooth, yet fast 10's and BIG, deep 15's. In my 15'x15' room, my Wharfedale W90's (in my avatar picture) have fantastic bass (Great dynamics and transient-response) and equally-fantastic mids and highs, and they can be found for well-under the OP's budget, but I have no idea what kind of sound he wants, aside from the big bass. They have two 12.5" woofers, two 5" mid-woofers, and two 4" Super 3 cone-tweeters per cab. They put out a smooth, deep, lifelike bass with good, usable output down to at least 30hz, plus no upper-bass boom (No boosted bass to make it sound bigger than life... Just whatever is on the recording). And Bozaks, as someone suggested earlier, also have fantastic bass and similar-sounding mids and highs (And the Wharfedale W90's AND the larger Bozak models are quite-big... VERY-large cabinets, even on the Bozak B-302a). But these are laid-back-sounding speakers which might not be what the OP is looking for here. They're a "You are there" type of speaker.. Very-detailed and dynamic, but also very-smooth and relaxed-sounding, which is just the way I like it. Some people prefer a modern, studio-monitor sound in a home hi-fi speaker. That's not really for me, but a lot of people like it. But whatever the OP chooses should be something rich, smooth, and well-balanced. Big bass won't matter much if the mids and highs sound like ass...