After the agent installation, it will be started as the "Local System" account by default (which has the high class of permission). In some cases it is necessary to reduce the risk of abuse.
I've tried to start Zabbix agent on a Windows server as LocalService, NetworkService, User, Power User, Performance Monitor Users and Administrator. I was able to start it as the Administrator user only.

Error message during the start and error from the event log is added:

- System 
  - Provider 
   [ Name]  Service Control Manager 
   [ Guid]  {555908d1-a6d7-4695-8e1e-26931d2012f4} 
   [ EventSourceName]  Service Control Manager 
  - EventID 7034 
   [ Qualifiers]  49152 
   Version 0 
   Level 2 
   Task 0 
   Opcode 0 
   Keywords 0x8080000000000000 
  - TimeCreated 
   [ SystemTime]  2014-10-31T07:46:12.211433100Z 
   EventRecordID 6075 
   Correlation 
  - Execution 
   [ ProcessID]  484 
   [ ThreadID]  572 
   Channel System 
   Computer W...A 
   Security 
 - EventData 
  param1 Zabbix Agent 
  param2 3 

It would be great to have a description of the minimal permissions which need to start the agent.