In Firefox 3, the answer is 6 per domain: as soon as a 7th XmlHttpRequest (on any tab) to the same domain is fired, it is queued until one of the other 6 finish.
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With IE6 / IE7 one can tweak the number of concurrent requests in the registry. Here's how to set it to four each.
[HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Internet Settings] "MaxConnectionsPerServer"=dword:00000004 "MaxConnectionsPer1_0Server"=dword:00000004