How many switches can I connect together on the same LAN?

Switches can be cascaded almost without limit because unlike a passive hub that has retransmission limitations, an active switch regenerates Ethernet messages as if it were an original transmission. 

There really is no limit as long as the total latency is acceptable to your application. For example, the total latency of 10 Ring Switches cascaded (connected in series) is typically less than 50 us. 

Content réference
Ethernet Military Switch RES-SCE-AC-8US
Ethernet Military Switch RES-SCE-8MG
Ethernet Military Switch RESMLAC-8US-CAPS
Ethernet Military Switch RESMLAC-8MG-CAPS
Ethernet Military Switch RESMLAC-8MG-CAPS F35
Ethernet Military Switch RES-SCE-12MG
Ethernet Military Switch RESMLAC-28MG
Ethernet Military Switch RJSMLAC-8UG-CAPS
Ethernet Military Switch RJSMLAC-8MG-CAPS
Ethernet Military Switch RJSML-MG7F3G
Ethernet Military Switch RJSML-8US1/8UG1