If you try a new battery, ensure it is NOT a calcium/silver-calcium variety. Your alternator is NOT capable of charging this type, try a Halfords HB096, it's a traditional lead-acid battery that uses lead-antimony as a coating on the plates to help prevent sulphation. Newer batteries use silver-calcium which increases the necessary charging voltage to 14.7V MINIMUM just to start the charging process!
In fact, if your current battery is a calcium variety, this may be your problem.......
https://en.wikipedia.org/wiki/Silver...alcium_battery
Have a read of that, you'll see what i mean. Also do some more research as you will find that 14.4-14.8V is a conservative estimate, it's normally 15.1-14.7V for a calcium battery but the point it does make in that Wiki page is that calcium batteries do deteriorate rapidly if they're not being fully charged at the correct voltage.