If you mean servers aren't allowed, then I suppose my heaviest desktop machine would be my iMac G3. Not fiercely impressive in weight, but CRTs aren't light either.
If servers /are/ allowed, then my IBM eServer 235 would win the heavy metal box game within my collection. It's a 4U server with dual 600W PSUs, Ultra320 SCA-80 hotswap bays for up to 6 drives (max capacity 300GB each). It's honestly my neatest system, with dual IA-32 P4-era Xeons, real PAE support up to 12GB of RAM, dual RS-485 and gigabit ethernet, lotsa PCI-X, and its most interesting party trick, hot-swappable PCI-X (which I believe not only requires the slot/controller to support it, but the card and OS to support it as well, restricting it primarily to NICs and maybe HBAs on Server 2003).