Charging and Using a Li-Po battery simultaneously

What you're describing is an "always-plugged-in" device with a "backup" battery. In addition, the "backup" battery may be able to provide significantly more current than the built-in power source, for short cycles, and when the load is removed, the battery is charged back up.

LiPo batteries don't like staying at top voltage (4.2V rated, typically) "trickle charging," because this will metalize the lithium, which will kill the battery.

However, it is safe to "float" a lithium polymer cell at a lower voltage -- typically somewhere between 3.9V and 4.05V, depending on the manufacturer and cell specifics. Thus, it is totally safe to design a cell "charge/float" circuit that provides a float voltage that won't go above the safe float voltage. This circuit will basically provide no current into the battery when there is no load. When there is load applied, the power will come from the battery and from the "float" voltage in relation to their respective source impedances (internal resistances.)

If the use is for something like an "always ready flashlight" that can also be on while plugged in, yes, you CAN build one of those with a LiPo (or a LeFePO4, which is a little safer, but has lower voltage per cell) by floating the cell at the appropriate voltage. You have to design the source impedance of the charger/floater to have the appropriate characteristics for the case of a heavy load on the output, versus charging a discharged cell, versus there being no load and a "full" cell.