There is no specific limit.
On a modern motherboard you have PCI-e slots and optionally a few build in Ethernet ports.
In each PCI-e slot you can add an expansion cards. That can be a traditional cheap NIC with one port, or more server orientated cards. Many of these feature 4 ports.
Thus, without doing anything exotic:
- Count the number of ports already on the motherboard.
- Count the number of free PCI-e slots.
- Calculate first + 4x second.
As to exotic: The can use PCI-e to multiple PCI-2 bridges.
Thus far the generic answer. Now for your specific setup:
Your board has one on board NIC, three PCI-e conectors and two ancient PCI-connectors. In theory you could have up to 21 Ethernet ports.
That is assuming no other cards are in use, and using PCI rather than PCI-e is not recommended (it is a shared bus with relative low bandwith, you you can put a quad card in it, but a single 1Gbit NIC can saturate a 33Mhz/32bit PCI bus).