I was just wondering what the maximum number of down votes a question can have before it is removed from StackOverflow.
According to the data explorer the SCORE column of the POSTS table is an INT.
Stack Exchange uses SQL Server, the minimum value of an int is -2,147,483,648, so this (unless there's a table with a smaller datatype somewhere).
As Toomai mentions this is the minimum possible score. As score is upvotes - downvotes the maximum number of downvotes is actually double this (assuming you also got 2,147,483,648 upvotes).
As TronicZomB points out it's also constrained by the number of people with the downvote privilege. The number as of the last dump was 175,988 so this was the theoretical maximum on 2013/06/25. It will be higher by now.
If by "removed" you mean removed from the system entirely, there is no number of downvotes that will result in a question being removed. Deletion can be done by the question owner, one moderator, or a group of community members who have enough rep to vote to delete. However deleted questions are not removed and can be seen by members who have enough rep to see deleted questions.
If by "removed" you mean not shown on the front page, it's -9 on Meta and I believe -5 on StackOverflow. See On Meta, are the rules for showing on the main page different?