I have a cell phone that has a 1500 mAh 3.7 V battery. It comes with a 700 mA charger but I've sucessfully used a 1 A charger with no problems. I'm now trying to make my own multi-device charging station and my first problem if figuring out the maximum amount of current I can use at 5V?
If you are charging the battery through the phone then this will have the charge controller circuitry between the 5V charge supply and the battery. You CANNOT/MUST NOT just connect a battery pack to a power supply and expect it to charge without fire and or explosion.
The charge controller in the phone will limit the current supplied to the battery pack to be within the limits specified by the battery manufacturer to ensure that the battery is not damaged. Supplying the phone from a 5V source that has a higher current capability will not make the battery charge any faster. If it did then you would run the risk of damaging the connector on the phone or even melting the tracks on the PCB within the phone. Small USB connectors that I have used have a contact rating of up to 1A on the power lines.
If you want to build a charging station to charge multiple phones at a time then you need to have a power source that can supply up to the maximum charge current taken by the phone down each of the charging leads. These can all be in parallel but I would place a diode in each of the positive supply lines to prevent the possibility of any current flowing from the battery back to the charger.
It's probably 1C, which means 1xcapacity, or 1500mA. However, there is a reason the manufacturer chose 700mA. It's best to keep to that to preserve battery lifetime and for safety. Overcharging a lipo can damage it. It may, in rare circumstances, become "puffy", or even overheat or burn. So be careful!
Never charge a lipo battery without a proper charger. They must not be exposed to a charging voltage exceeding 4.2V. They should be charged with a constant current and monitored for voltage. Never connect a lipo directly to a supply.
Current is not limited as long as battery does not overheat internally & externally. You can have higher currents with active cooling.
Going above 1C is possible (i.e. 1.5A in your case) but it will reduce lifespan of your battery.
Under no conditions you should connect unregulated 5V to LiIon - current will be >10A and something would explode (PSU or battery). You need regulated current circuit.
It sounds like what you're building is a dock that will accept multiple phones and power them all at the same time. A block of plastic with a micro-USB connector sticking up for each phone?
If every phone has a wall adapter that puts out 5V, or has a USB connector that accepts standard USB output (which is 5V at 0.5A), you can add up all the currents and make sure the power adapter for your dock can supply at least that much current at 5V. It's fine if it can supply more current, because the phones will only take as much current as they require. The official USB spec (thus most ports) only offers 0.5A, although some manufacturers give the device more current when connected to the wall adapter, notably the iPad. Making the power adapter unnecessarily large does add some extra risk in the case of a mishap like an accidental short circuit.
A fuse on each line might be a good idea. Ian suggested adding a diode between the main power supply and each phone's connector, but I'd be wary of that because 5V - 0.7V = 4.3V, which does not leave a lot of room for the phone's internal battery charger to step down to the 4.2V the battery will require to be fully charged. The phone is expecting to see 5V, not 4.3V, and is designed accordingly.
If you are only powering the phones and not the batteries directly (removed from the phones), you don't need to worry about 4.2V, 1C, or how the battery is being charged. The phone will handle that for you.