Is it possible to build a non-degrading battery?
But is it possible to have a battery that will never degrade and always keep its performance without upkeep/maintenance?


Well, consider the dry pile powering the Oxford Electric Bell, which has been running since 1840. However, that is an example of a nonrechargeable pile.

The question is, can a rechargeable battery be designed that retains it's charging/discharging current profile, reflecting a conserved structural integrity, during an unlimited number of cycles?

While in practice there are many reasons this is difficult to do (loss of electrolyte, occurrence of irreversible passivating reactions, etc) in principle there seems to be no reason why this should not be the case, at least over time on the order of a human lifetime. We manufacture many things which last an awful long time, and batteries would last even longer if designed or treated in an exceptional way, shielded from the environment and charged/discharged in a highly regulated manner. Some battery designs such as thin film lithium-ion allow an exceptional amount of cycling. 


  I just didn't know if there were some types of laws in chemistry that prevents this from happening.


The second law of thermodynamics places limits on what you can expect to achieve given the constraints of a particular choice of battery chemistry. This reveals itself for instance in Peukert's law, particularly applicable to lead-acid batteries operating at constant temperature, which says that the capacity of a battery is restricted by its discharge rate. Discharging quickly reduces the total work you can do with the battery.  

The second law however does not say we cannot use work to recharge a battery to the same original state a very large number of times. But it does make it very hard! It begs the question, "is such a  battery useful?" Probably not, except theoretically. The design and operation of a battery involves tradeoffs between charge density, discharge and recharging rate, and preservation of structural integrity during charging/discharging cycles.

By the way, you could generalize the question to include other portable high-capacity sources of electrical power. Think of probes that are sent into outer space and continue to operate after, well, a long time. Or consider the quantum battery.