Lithium Ion battery stopped charging

(the numbers are given for LiCoO2 chemistry, for LiFePO4 they should be lower) Short answer: yes, this is OK. I charge lithium cells with a bench supply set to 4.2 V and whatever C/10 figure is for the cell I'm charging. If you are not in a hurry, slow charging is better, even for cells claimed to be capable of withstanding higher charging currents.

However (quoting you):

charging at a constant voltage (say 4.2V) so long as the maximum current is limited to a reasonable value for the cell

means you will have constant current charger till your cell is at ~95%. Up to this point the voltage across the battery will be less than 4.2V if you measure it. Only when your charger starts outputting 4.2V it will become constant voltage.

What you are about to build is CC/CV charger and this is the right thing to do. "Constant voltage only" charger will be set to 4.2 V with no current limiting and it will charge the lithium cell very slowly. You can check it youself, just construct var.voltage circuit and measure the current into (discharged) cell at 3.5, 3.7, 4.0, 4.2, and 4.5 V. Cheap Chinese chargers are constructed like that, they restrict the voltage to 4.2 V so the cell won't ignite after being charged but the consumer would have to wait longer. I once bought a portable emergency charger which could be emptied in a couple of hours then took 3 days to recharge its internal cells.

There are other precautions to observe while charging lithium, you can learn them from any modern charge controller IC datasheet (my favorite is Linear Tech., their literature is very high quality). If you don't implement these precautions in your design never leave it unattended while charging, otherwise it may ruin your morning one day.

Lots of questions! Instead of just pointing you to a certain IC or battery, I will try to answer your questions so that you can understand the concepts and pick or build your own battery.

1. What precautions do I need to take so that there are no explosions? Certainly something that is possible with lithium ion batteries!

   A) Do not overheat. You can find charging systems with thermistors that monitor the temperature of your battery.

   B) Do not over discharge. Lithium ion should not be discharged below 3 volts, but 3.3 volts is a better goal to aim for. You Over discharging is more likely to cause the battery to lose capacity than explode, but best to keep over 3 volts regardless.

   C) Do not over charge. Very important not to charge over 4.2 volts.

   D) Do not short circuit. Allowing the positive of the battery to come in contact with the negative for longer than a few milliseconds will likely cause a fire. Usually the short circuits are accidental and there are circuits that will protect the battery if short circuit occurs

   E) Do not discharge the battery at a rate faster than the battery is rated for. Again there exist protection cicuirts that will protect against this.

   F) Cells in series must be balanced. Simply applying a current to the battery pack might result in some of the cells charging at slightly different rates leading to imbalance. As a result it is a very good idea to monitor each cell as it is being charged. There exist IC that will do this.

2. How does the use of boost and buck converters affect battery life? Boost and buck converters are quite useful, but you will lose some battery life due inefficiencies. Most of the boost and buck converters that I have seen have about an 85% efficiency although occasionally I have seen 95% efficiency claimed. Thus you lose 5-15% battery life due to inefficient conversion.

Don't forget that a fully charged 3 cell lithium ion battery is 12.6 volts - is this too high?

Good luck.