memory is usually the one that gets clobbered. Other than that, depending on what you're doing, i'd say hard drive space. If you need it fast then concentrate on CPU, but otherwise, it'll get done, just a bit slower
There's no way to really know what will run out first as it'll depend on whats hosted. If you're running CPU/Memory intensive php scripts or forums, then those go first. If you're hosting lots of files or disk space intensive sites, that'll of course go first. Memory is pretty much guaranteed to scale up in use regardless of what you're hosting as you add more sites, scripts, and linux starts caching more data.
At a minimum I wouldn't get less than a 80GB drive and 2GB of memory for a starter server. If you're going with IDE drives you can always scale up the storage to 160GB with minimal cost. You could get away with 1Gb of memory, and add 2 more 512mb or a 1gb stick later though.
Just try to take your best guess based upon your plans and be ready to adapt should it be a sucess and you need to expand.
Originally posted by RazorBlue - Dan I have to agree with James, people often think that their server is slow so they'll upgrade the CPU, when it's really the RAM that plays a massive part in server performance.
Or the disk I/O. There are some applications the will benefit greatly from using SCSI drives.