Hmm, let's get a baseline. Assume an average tribble is a sphere with a radius of 5 cm, therefore a volume of 0.00052 m^3. The most efficient packing of spheres takes up 74% of available volume, so let's multiply that by 4/3 and call it 0.0007 m^3. So 1,771,561 tribbles -- three days' worth of reproduction -- would take up a volume of only 1236.8 cubic meters, equivalent to a cube 10.73 meters on a side -- or maybe about half the residential volume of the 15-unit apartment block where I live. Even a small K-7 could hold a lot more tribbles than that.
So how big is the Death Star? Estimates of the first one's diameter range from 120 to 140 km, which makes for a lot of variation in the volume estimates. But taking a roughly midrange value, the volume would be somewhere around 10^15 m^3. Therefore it could hold around 1.4 x 10^18 or 1.4 quintillion tribbles. (That's 1.4 billion billion.) Of course, realistically the tribbles would run out of food and starve to death long before they reached that number.