PROBLEM STATEMENT
You are running the Office 365 / Directory Synchronization tool with a SQL Server Express backend. You have less than 50,000 users. Now, you are seeing information in the Application Event Log indicating that the database is full.
Additionally, you may receive Event ID 1827: Result cumulative size will exceed license limit from 402 to 10,000 mega bytes, in the Application Event Log.
CAUSE
The reason this happens, is because each run is logged on the Operations tab of the MIISCLIENT.EXE console. This is commonly referred to as the Run History. If there is no process to clean up the run history, then it will continue to grow, and the MDF file of the SQL Server Express database will grow as well until you reach the 10GB limit.
RESOLUTION
To resolve the issue, you will need to clear the run history. To prevent the issue from happening, you may need to build a script that can be called from Task Scheduler to clear the run history.
Here is a Microsoft TechNet Wiki that explains about clearing the run history.
Tim Macaulay edited Revision 2. Comment: added the event id information to the page