What happens if we re-charge a Lithium-Ion battery when it is not fully used?

No, the battery will handle that fine.

.

it will develop a memory, and only hold that much of a charge...

Example..

Your battery when new lasts 2 hours.

You use it for an hour...

you charge it...

Now the battery thinks that when fully charged, it only lasts an hour...

Why, I do not know, just know what happens...

Lithium-ion technology does not suffer from the "memory effect" that some of us remember from the old Nickel-cadmium batteries.  LI will handle a partial charge-discharge cycle without any long term problems.

This should be fine.  Li-Ion batteries may suffer from repeated deep cycles (completely discharging then completely charging), although I'm not 100% sure.  Anything else should be fine, though.

Doushebag is likely wrong. Memory is a peculiarity of nickle-cadmium batteries. Most other types of batteries do better with frequent slow charge. Lithium ion batteries are better in almost all ways. Disadvantages are initial cost, lithium ion batteries deteriorate if they are excessively discharged below about 3 volts, or excessively charged over about 3.7 volts, per cell. Computers can solve these problems before the deterioration begins, but experienced humans can coax some more usable life out of partially deteriorated lithium ion batteries.    Neil