I really like hybrid cars e.g. Toyota Prius ... but I would like to drive them without their battery so that I can drive them only on petrol and don't have to worry about battery maintenance / replacement. Is it possible that I remove the battery from Toyota Prius and drive it like a regular non-hybrid car?
No, this is not possible.
The high-voltage (HV) battery is used to start the engine. Remove this battery, and you soon find yourself unable to start the engine. The engine is started by motor-generators (MGs) operating on the HV battery, not a conventional starter motor operating on 12V auxiliary battery.
Also, the simulated Atkinson cycle engine would totally suck without the electric boost from the HV battery. So if you somehow managed to install a traditional starter motor to a Prius (which ain't easy), you wouldn't like the car at all.
You are worrying about the wrong things. The battery is reasonably durable. Consumer Reports has tested two about 10 year old Prius cars, one had only few thousand miles and the other a lot more. Both had good fuel economy even with 10 years and for the other car, lots of miles. If you find yourself with a bad battery, a junkyard battery can probably be found and installed for less than $1000.
In case you have a bad battery (which can happen, as evidenced by a comment here: Toyota Auris Hybrid sometimes is reluctant to enter READY mode), in most areas Toyota has a good warranty for the battery. In Finland, the warranty is 10 years or 350 000 km.
It's not possible - and it's probably not what you want anyway. My first Prius lasted over 350,000 miles (570,000 km) without any battery maintenance whatsoever. If there were a way to run without the battery, you would be making the petrol engine do much more work, increasing your maintenance cost.
Toyota HSD is careful to avoid over-charging or depletion of the HV battery (the battery gauge on the dash actually ranges from 40% to 80%, rather than the 0-100% that it looks like), so battery life is much better than devices like cameras that you tend to fully discharge.