General:
* Complete
* Fulfill
* Satisfy
* Pack
* Stuff
* Load
* Occupy
Specific to Filling a Container:
* Populate
* Replenish
* Top off
* Refill
* Charge (for liquids)
* Pour
* Infill
Specific to Filling a Form:
* Complete
* Submit
* Enter
* Provide
* Compose
Specific to Filling a Role:
* Occupy
* Assume
* Take on
* Hold
Specific to Filling a Gap:
* Bridge
* Close
* Cover
The best synonym depends on the specific meaning of "fill" in your context.