The charging light on my Medion laptop goes out when the battery is fully charged, even though the power pack is plugged into the computer and switched on. I presume that means there is no trickle charge happening, and therefore no risk of overcharging, though I don't know for sure. The manual doesn't say much in that regard.
It does say that, to keep the battery conditioned, it should be run right down and then recharged. That seems to be common to most rechargeables, except lead-acid which don't like to be fully discharged and are not used in computers anyway.
Maybe you should consider the discharge-recharge cycle, or ask your service rep.
Cheers
AA