Powering up STM32L from Lithium Ion battery
Please read carefully all data sheets.

(1) The STM32L100RC MCU discovery Discovery board, per its user manual, has embedded LDO from 5 V to 3.3 V, minus Schottky diode D2 for power rail decoupling, which makes the output at about +3.0 V, and it is labeled as such:



As per this design, the Discovery kit can be powered at least in two ways:

 (a) through the standard USB port (5V 500 mA), or 

 (b) directly by 3 V supply, bypassing the LDO U1. 


Obviously the whole system has wide margin, which doesn't mean you need to stress all option and overvoltage the board.

(2) Regarding the 18650 Li-Ion battery, your conclusion that "it seems that it can't ever output more than 5V, and over 200mA," is staggering.

While yes, Li-Ion batteries never output more than 4.35 V, the datasheet for this particular battery clearly presents data that it can be safely discharged at 2C rate, meaning that you can get up to 6.4 A of current, easily, and not just "200 mA"

Furthermore, the Discovery kit uses on-board LDO regulator LD3985, which can output only up to 150 mA, which means that the MCU cannot consume more that that. And the input voltage range is from 6 V (absolute max) down to 2.5 V. Which means that you don't need any additional "LDO reguator" and can safely feed the board either from your undisclosed "PV solar panel" (assuming its output is fixed at 5 V), or feed the +5 V power pin directly from a charged NCR18650B battery. Be aware however that the LD3985 doesn't have "undervoltage" protection, and its parameters are not guaranteed if input is below 2.5V. Therefore some special measures should be taken to prevent the cell overdischarge since the NCR18650B  cell is unprotected.