What is the maximum current which can pass in a Li_ion battery?
As a rule of thumb small li-ion or li-poly batteries can be charged and discharged at around 1C.  "C" is a unit of measure for current equal to the cell capacity divided by one hour; so for a 200mAh battery, 1C is 200mA.  

Example: common 402025 150mAh battery from Adafruit: quick charge 1C, maximum continuous discharge 1C.  

Slower charge and discharge eg 0.5C or 0.2C gives better capacity, close to the nominal for the battery, as well as longer life in cycles.  Many battery datasheets only guarantee the number of cycles for 0.2C charge, even though they do allow up to 1C charge.

Individual batteries differ a lot; some I've seen for example only allow charging at 0.5C max in the spec sheet (which is pretty slow for a battery made nowadays), and others such as quadcopter batteries allow discharging at 10C or even 20C (which is impressive, but also potentially very unsafe...)  So the rule of thumb is definitely not perfect; check the datasheet. 

P.S. This is for small batteries under 2000mAh or so; large batteries are a different game, party because they are often packs made of small batteries in series and/or parallel, and with much more complicated management circuits.