Lithium batteries folks
For extra paranoia, lithium batteries are a fire risk when they aren't charging, too. That's why you will have noticed on recent plane flights that the safety video now includes "if you drop your phone into the seat, please call cabin crew immediately", that a lot of airlines won't let you check those silly "smart bags" with built-in batteries and there is a lot of nervousness about lithium cells in the hold more generally. If you short out a lithium battery, which can also happen just by bending it, then it's a massive fire risk: lots of charge density, and the battery will burn very hot, too.
It's not at all obvious that lithium batteries are significantly more likely to catch fire while charging than otherwise.
www.thoughtco.com/why-lithium-batteries-catch-fire-606814
"Over charging" isn't a risk in any well-designed device: modern equipment assumes that the charger is shit (ie, that it might provide roughly the rated voltage, at roughly the rated current, but might not) and shuts down charging either on reaching capacity or on the charger mis-behaving. Indeed, one thing that's a pain is you can't power a phone from a power pack: once it's charged, the phone disconnects, and the phone's battery is then used.
Unfortunately, some laptops aren't "well-designed", and rely on the smarts for dealing with the battery being in the charger, not the laptop. That's going to go away with USB-C, because there the charger doesn't know the laptop from any other load, but for a lot of older laptops, charging with a replacement charger is less safe, because it continues to charge the battery even when it is full.
But this started out with a discussion about phone chargers. Has anyone got a phone in 2018 where the charger isn't a USB connection of some sort? In which case, all the charge protection circuitry has to be in the phone.