Most motherboards have a pair of IDE channels on them, two drives per. You can add at least two additional IDE controller cards (most I've done, more may be possible) for an additional four drives per-- that's ten IDE drives.
It is possible to RAID a bunch of smaller drives into a single virtual volume. Each SATA channel holds one drive, though one can use a port multiplier with SATA II for up to 15 drives per channel (bandwidth is a limiting factor).
I have yet to see anything that specifies a technical limit to the number of drives one can attach to a computer, but common sense dictates that physical architecture and available bandwidth would impose limits.