Because your button is located outdoors, it is subject to all the elements. Sometimes the plastic doorbell button cannot withstand the temperature changes, wind and dust that come its way. As a result, it starts to warp or jam so that users cannot push the button all the way in. Unscrew the button assembly and look inside to see if it needs to be cleaned. If the button itself is too damaged to meet its electrical contacts, you will need to replace it.
Doorbells usually have two terminal connections that the button must touch together to have the doorbell send the proper signal to the inside system. Use an ohmmeter or simply connect the leads to the button manually, after unscrewing the button to test it, and make sure it works. If it does not work, you may have a shorted wire or circuitry that needs to be replaced.
If your terminals look especially dirty, try cleaning them before you give up entirely. Sometimes terminals can rust or collect grease and grime over time, until the circuit cannot be completed when the doorbell button is pushed. Clean the terminals and leads thoroughly, then try using your doorbell again to make sure the system is actually damaged and not just dirty.
Do not assume the problem must be with the doorbell button itself, if the doorbell doesn't work. Sometimes an electrical short can occur in the transformer or even in the soundbox itself. If the doorbell appears to be working correctly, move onto these other systems.