Having blown a PiModules UPS Pico I played around with the LiPo battery 450mAh as it has an embedded PCB (Over Charge, Over Discharge, Over Current) inside. After connecting it directly to pins 4 & 6 of the RaPi, I turned on the AC power supply and measured the current into the LiPo & its voltage across it while checking for overheating, etc. There was no heating & 25 minutes later it was charged at 4.2V with the current starting at 1.5 (3C !) & dropping to 0.9A (2C).
On disconnecting the AC the RaPi stayed on with the small, multicoloured image indicating low power showing from the start. With a steady current of 0.7A the RaPi was functional for 8 minutes, with the mouse going slow at 6 mins. The keyboard stopped working at 10 mins. At 20 mins the screen went blank but when I plugged the AC in all the open files reappeared.
I understand that a few LiPos have exploded but was that before the embedded PCB was added? I have read that they can be charged/discharged between 100 & 500 times and am tempted to test this battery in this way. I would set up a thermistor next to the LiPo &, if the temperature went to say 30°C, would shut down the experiment automatically.
If the experiment worked it would make a very cheap UPS system!
See the comments I have made about the impartiality of @laptop2d below her or his answer before dismissing my idea out of hand.
14 June - At the moment I am working on the idea that the LiPo will register that the 5v power supply has failed & instruct the RaPi to use the 'sudo shutdown -h now' instruction. I am still try to find an instruction that will save & shutdown any open documents like LibreOffice, automatically. The LiPo will be used for, say, 30 seconds before being cut-off from the RaPi automatically with a timer circuit & then recharged for a short time when the 5V power supply restarts. At this point the LiPo voltage can be monitored by an ADC like the MCP3008. This would be a useful add-on for a RaPi.