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The hard-and-fast limit to the number of VMs you can run is your computer's memory. However, the more virtual machines you have, the slower everything will get - in particular, disk I/O is typically the first and worst performance bottleneck you'll face, since most disk performance degrades much worse than linearly with increased concurrency.
The number of VMs you'll be able to support will depend on several factors – the capacity of the server hardware, the efficiency of the hypervisor, and the requirements of the guest operating systems. Server hardware can support up to four 12-core processors, 256GB of RAM, and four or more quad-port Gigabit Ethernet or dual 10G Ethernet adapters, along with enough high-speed storage for dozens of VMs per server.
Efficiency relates to the ability of the hypervisor to recover unused resources when a VM is not in use. CPU resources and memory can be allocated only when needed. Storage space can also be thin-provisioned, which means that even though a VM has an 80GB virtual drive, only the 10GB or so actually in use to store files will occupy space on the storage system, rather than the full 80GB.
Of course, guest operating systems will vary widely in their requirements for CPU power, memory and storage. For example a Linux server might only need half a CPU core, 512MB of RAM and an 8GB virtual disk, while running Windows 7 optimally will require at least one CPU core, 2GB of RAM (preferably four), and 20GB or more of virtual disk. Not all Windows guests will require lots of resources – Windows Web Server 2008 R2 can operate with minimal resources. In general though, Linux will require less in the way of resources than other OSes.