What battery should i use for a bluetooth boombox?
The only chemistry that I can think of that's relatively safe, and in fact what I've chosen for my portable boombox, is SLA, sealed lead acid.

Charged at the float voltage, 2.3v per cell, or 13.8v for a 12v battery, it will accept that voltage indefinitely, without degrading significantly. The slight overcharge current is tolerated by the gases recombining within the cell. This process has a limited rate, so the maximum float voltage (to limit the overcharge current to what can be absorbed) must be strictly observed. 

The maximum charge current needs to be limited as well, to protect it when it goes back on charge after a lot of use.

Do be aware that lead acid does not like deep discharge cycles. If you limit discharge depth to 50%, you will get an order of magnitude more life out of the battery than if you allow 100% discharges. 

These characteristics suit my style of use quite well. It can be on float charge permanently when at home. The 'normal' run time is at the 50% capacity rating, and if stuck away from home, the extra 50% capacity is regarded as an emergency reserve, that I can use occasionally.

I discounted VRLA (valve regulated) as they tolerate overcharge by venting, and have to be topped up with distilled water from time to time.