you could try 'show snmp', normally the first line shows the SN as well.
I wonder why show inventory doesn't work. What IOS version are you running?
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I also wonder why show inventory doesn't work, but if you do a show ver, the Processor board ID is the Chassis serial number...at least it matches mine.
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That's wierd. I always use "sh inventory raw" and the first line is the chassis' serial number. If not, then if you haven SNMP enabled, use the command "sh snmp" and it'll come up as well.