A lead acid battery seeing 13.8 volts at the terminals means the alternator is doing it's part. That's actually at kind of the high end of what you would see. If you are anything over like 12.6 volts the alternator is supplying an excess of current for the car and the battery is charging. One could argue the alternator is weak and the battery is not charging fast enough, and that's possible, but if the battery could start the car and then is charging (even a little) then the battery should generally start the car again most times, so the problem is likely not the alternator.
I'd also respectfully disagree with the "a car can't charge a flat battery" statement. An alternator needs some battery voltage to bootstrap the whole power generation process (to create the magnetic field that the motor then forces a coil through to generate current). But even a little is enough to create enough of a field to create enough current to create more of a field to create more current... etc. So even a little current will be enough convert the ample mechanical energy of the internal combustion motor into a "net gain" current that will both give the alternator more magnetic field than it needs and to also start charging the battery.
And the alternator itself regulates its power output depending on required draw, and likely has a 100 watt to 300 watt margin of "extra capacity" under normal running operating conditions. For a 13 volt system, even with just 100 watt capacity to spare, that's 7 amps of available current for battery charging. A car battery is something like 50 amp hours, so in 7 hours of driving it should be able to comfortably completely charge the battery (worst case). It probably has more like a 200 watt margin and so could do it in 3.5 hours, which is probably about as fast as you would ever want to completely recharge the battery anyway to avoid frying it.
So anyway, while alternators do fail (mine did after I did my U-Boat commander stunt), its usually the battery or connections between the battery and car or battery and starter. And once you see 13.8 volts on a running car at the (clean and tight) battery terminals, it's likely a battery problem.
It could be a parasitic drain problem, but a 50 amp hour battery on a .1 amp load would take 500 hours to completely discharge, so 20 days. .1 amp draw (if it is really that, a cheap meter might do fine with volts but be sketchy for amps, and amps are harder to measure) does seem pretty high, that's enough to run a dome light, but if you are talking a car not starting after sitting overnight, the problem isn't a .1 amp draw. Though it could be a .1 amp draw combined with a "mostly dead" battery.
Batteries can and do fail quickly sometimes. They don't seem to be made as well as they used to be made.
So anyway, it smells more like a battery or battery connection problem to me from what you have observed so far.
(Sorry for the long rant... I'm trying to pay back help I'm getting here on other threads)