How many primary partitions can be created on a computer running Windows 8?

Strictly speaking, Windows only ever allows 4 primary partitions, because the phrase "primary partition" is only applicable to MBR partition tables and MBR partition tables only allow four primary partitions.

However, if you use a GPT partition table (available since Windows Server 2003 on 64-bit versions of Windows) you can have up to 128 partitions and do not need to distinguish between "primary" and "extended" partitions.

Depending on your computer model, you might not have the option of using a GPT partition table on your boot disk, because Windows only has native support for booting GPT disks via UEFI, not via BIOS.

What changed with Windows 8 is that computers sold with Windows 8 must (in order to be certified) support UEFI booting. They must also ship configured to UEFI boot; as a consequence of this they must be shipped with GPT partition tables.

The number of primary partitions isn't a limitation of the operating system, but of the partition table.

• If you use an MS-DOS partition table, you can have up to four primary/extended partitions.

• If you use a GUID partition table with default settings, you can have up to 128 primary partitions.

gptgen is able to convert an MS-DOS partition table into a GPT:

Gptgen is a tool to non-destructively convert hard disks partitioned in the common, "MSDOS-style" MBR scheme (including extended partitions) to use a GUID partition table (GPT).

I haven't tried the tool myself, but I found a few success stories on Google. Keep in mind that any modification of the partition table carries the risk of data loss.