The short answer is "about 4 to 6 connections per domain"
For more details, check out this great roundup of how many parallel connections each browser will make to a given domain
Results summarized below...
+----------------------+------------+------------+
|Browser | HTTP/1.1 | HTTP/1.0 |
+----------------------+------------+------------+
|IE 6,7 | 2 ## | 4 #### |
|IE 8 | 6 ###### | 6 ###### |
|Firefox 2 | 2 ## | 8 ######## |
|Firefox 3 | 6 ###### | 6 ###### |
|Safari 3,4 | 4 #### | 4 #### |
|Chrome 1,2 | 6 ###### | ? |
|Chrome 3 | 4 #### | 4 #### |
|Chrome 4+ | 6 ###### | ? |
|iPhone 2 | 4 #### | ? |
|iPhone 3 | 6 ###### | ? |
|iPhone 4 | 4 #### | ? |
|Opera 9.63,10.00alpha | 4 #### | 4 #### |
|Opera 10.51+ | 8 ######## | ? |
+----------------------+------------+------------+
