I had a customer, where the vCenter server was running out of disk space, and i turned out to be, the vCenter syslog Collector log file that was growing rapidly, and had grown to +30 GB.
The solution was to disable the debugging on the service in the configuration file: %PROGRAMDATA%\VMware\VMware Syslog Collector\vmconfig-syslog.xml
-<debug> <level>0</level> </debug>
Changed the level to 0.
VMware has an KB on this: http://kb.vmware.com/selfservice/microsites/search.do?language=en_US&cmd=displayKC&externalId=2094175