I've been hearing about them since 2003-ish from my cousin. I never learned what they were about until I heard Panic Attack and Constant Motion on Rock Band 2 in late 2008. I instantly enjoyed them, and early 2009 I purchased the Greatest Hit compilation. I wasn't fully into them yet -- I didn't get Black Clouds as soon as it came out, but I got it for my birthday in October that year, making it my first true DT album.
Afterwards, I purchased Octavarium, Systematic Chaos, Chaos in Motion, Images & Words, Awake, Scenes from a Memory, Score, Six Degrees, Train of Thought, Falling Into Infinity, When Dream and Day Unite, Metropolis 2000, Live Scenes From New York, Live at Budokan, Live at the Marquee, and Once in a LIVEtime, all in that order. SC, I&W, Awake, SDOIT, LSFNY, LAB, LATM, and OIALT were iTunes purchases. And then of course, I pre-ordered ADTOE.