The other answer failed to mention how long the battery typically lasts, mentioning only the warranty of the battery. Furthermore, the other answer failed to mention that there are actually two batteries in Prius, the LV battery and the HV battery.
The 12V LV battery is a lead-acid battery and has the same replacement interval than conventional car batteries.
The HV battery will last the lifetime of the car. The reason this works is that the electrical system keeps the battery in the middle of its state of charge. Most wear on batteries occur when they are nearly empty or nearly full. Another reason this works is that the HV battery is oversized for its job and will work very well even with capacity degradation. Some new Priuses will have a lithium ion battery, and the lifetime of this is unknown but probably Toyota would not choose technology with lower lifetime than existing technology.
You don't just to take my word for the battery lifetime. Consumer Reports have compared an approximately 10 year old 200 000 mile Prius with a low mileage Prius being approximately as old as the other car. The conclusion was that fuel efficiency was similar on both cars.
Overall, Prius is probably the car with the least worries during its ownership. Many recent automatic transmissions fail easily such as the DSG transmissions that Volkswagen uses. Parts in manual transmissions (e.g. synchros) and clutches can wear due to user error or even due to normal use. The Prius transmission has only one speed (but three axles so it works effectively like a CVT when used with electric motors). This is the simplest and most reliable kind of transmission you can find! The engine is oversized for the power it produces, meaning it is probably very reliable. No failing turbochargers like on many new cars. Indirect injection means that injectors are cheap to replace if they fail and also that there should be no carbon deposits on the intake valves. The regenerative braking should mean that brake pad replacement interval is longer than on conventional cars. No accessory belts on the Prius to fail or to require replacement. The water pump and air conditioning compressor are electric, meaning they don't rev at high speeds when you're revving the engine, making quick wear less likely. The motor-generators will last longer than conventional alternator.