If it is a school project - and the teacher is at least reasonably sane - then the three types are NiCd, NiMH and lead-acid. I.e. no Li-Ion batteries involved. And none of the rechargeable "primary" batteries that exists.

For NiCd, NiMH and lead-acid, you can't normally explode the batteries with bad charging. NiCd can stand reasonably high currents by just getting warm. NiMH will age a lot if running too hot. Lead-acid cells starts to boil but should either be open or have a release valve for pressure release. The issue with all lead-acid batteries is of course that it doesn't matter if the charger is good or bad - the user must still make sure to not have sparks close to a charging non-sealed battery since hydrogen + oxygen is not a nice combination - more than one car owner have found that out by connecting the clamps directly on the poles and then remove the clamps without first removing the charge current.

When Li-Ion gets involved, it is absolutely imperative that a battery is not polarity switched. And that a too depeted battery is not charged. Or that a battery is not charged too fully. If they do not explode, they will still burn with very intense heat. And lithium does not react well to water.