I'm getting a new ebike with a 48V hub motor and a 10.4Ah, 500W battery. I was wondering how many times can a battery be charged up before it is time for a new one? I expect average charge each time 10% to 80/90%
How many times can a battery pack be charged? Much depends on the build quality of the battery pack. This includes considerations such as cell quality, BMS (battery management system) sophistication, care used in matching cell internal resistances, assembly techniques, charger sophistication, and more. How the pack is treated after it's put into use also impacts effective life. As you've mentioned limiting charge/discharge levels will impact this, as will storage temps, charge and discharge currents, and level of charge when in storage.
Data from batteryuniversity.com reports up to 3x battery life when charge/discharge is limited to 20-80%. They also report that effective life can be significantly extended with charge currents that are limited to 0.5C and/or discharge currents that are limited to 1C where C(amps) = (battery capacity in WH)/(battery voltage).
With all things considered battery effective life can be highly variable. There are some reference points. For example Specialized offers a 2 year battery warranty assuring the user of a minimum 300 charge cycles with a minimum of 75% of initial battery capacity remaining. Specialized battery packs use quality cells and sophisticated BMS and charger designs. They expect a large percentage of their battery packs to last beyond 300 charge cycles. This can be extended by following the practices discussed above. YRMV.
BTW - I have a Specilaized ebike. At ~35 miles per charge in our hilly town I would need to ride 10,500 miles to get to the 75% capacity Specialized uses as an end of effective life definition. My plan is to buy a new battery as the capacity on the original drops and then use the original as an 'extender' for longer rides. So with my approach the battery never really dies, it just goes into partial retirement.
No definitive answers for your battery I know, but hopefully these are some useful thoughts that you can research further.