Uh-Oh. Maybe.
Check the water levels in the battery. It is possible it boiled over, and if the levels are down, you may indeed need a new battery. As well, maybe, a new converter. If this has happened, be sure to rinse all those white deposits thoroughly to prevent corrosion.
It is possible for a bad converter to overcharge a battery, in the process heating it up until the acid boils. This, needless to say, kills the battery.
If you have a volt meter, slap it across the terminals while the converter is on to find out what voltage it is applying to the battery. Converters are supposed to apply about 14.5 volts when the battery is in need of recharging, and taper that off to a little over 13 volts to maintain a fully charge battery.
Disconnect the negative (ground) wire from the battery, and measure the voltage again across just the battery. It should be 12.7 or so if the battery is fully charged.
Under normal circumstances, you should be able to disconnect the ground wire from the battery to isolate it from the converter when you leave the rig plugged in while storing it. If you do, you should reconnect it for a few hours once a month to keep the battery in tip top shape. Whatever you do, you need to make sure that the battery acid levels are kept topped up using distilled water.
Could also be corrosion on the battery terminal(s). At any auto parts store you can buy for a couple bucks treated felt pads that prevent corrosion. They come as a red-green pair, with one color for the positive and the other the negative terminals. I'd recommend them.