I have a laptop at home which cannot stand without external power supply.. I replaced the internal battery but still not ok. The root cause must be the motherboard.
It is very frustrated, when we remove the plug, to loose all the open sessions, time and so.
For some times, I was thinking to add, in serial with the power supply, a device which can keep the laptop under power for some minutes and as well for hours under standby.
This computer, like many others, need 19v / 3.5A external power.
The original idea (and wish from my family), is to have as small as possible added device. The first idea was to use one 18650 battery but a rough calculation showed that it will need 20A to provide the necessary 70W. So, I decided to use 2 batteries 18650, which make 10A max.

Mock up with modules concept:

Before to make a final product, I decided to use some modules which can be find on the market.
The basic principle is
1) a DC/DC, boost mode, from the 2 batteries to 18.5V, with 3.5A output.
2) a DC/DC, burst mode,  to convert the 19v to 10v for the charging function
3) A balance charger to be sure the 2 batteries are not overcharge during charging
4)  A constant current based on LM317 to charge the batteries
5) some OR with diodes to power the computer either from regular power supply or from the (1) DC/DC

Description of the modules:

1) After some research, one good candidate for a DC/Dc from one or 2 lithium cells is the LTC1871. It can start from 2.5V, Need an external MOSFET. I found a module, HW-317 V6 with display of voltage in or out.

2) DC/DC burst, to decrease the charger voltage from 19v to 9v around (to be adjusted in order to have exactly 4.15v after the constant current module. Here, I use a module based on LM2596 like this one: https://electropeak.com/lm2596

3) Balance charger is necessary to avoid overcharge one cell. Several option are possible but I tried with a new IC called ETA3000.. Instead of wasting overcharge in a resistor, it use it to charge the lower cells. Not easy to find a module with only one ETA3000 which can balance 2 cells.  

4) Constant current is based on the LM317, with a resistor to define the current. Here, I use 1.5 ohm, which is a current 1.2v/1.5 = 800mA. This LM317 is going to heat quite a lot. Need plan heatsink. 

5) There is a OR with 4A diodes Schottky SSB44 to provide the supply to the computer. The output of the module (1) need to be slightly lower than the Ac/DC original power. Also, the LM317 to charge the batteries does it through another diode, to avoid the batteries to be discharged in the LM during standby.

Several capacitors, 1000uF / 25V have been added on the input power supply side and at output of the OR diodes before going to the computer.

For the batteries, I used Panasonic NCR18650GA which are rated 10A. I tried to find some LG rated 20A and check if it can work with one cell but cannot find them yet.

IIf the family is happy with this module, will make a full integrated module. So far, with this current mockup, some function need to be added: 
- Stop the Dc/Dc Boost (1) if the batteries level is too low
- Reduce standby current if possible
- Display current in and out

Please note that on the uploaded pictures, the ETA3000 module is not yet mount.

Laurent