MacBook Pro 2013 doesn't charge or boot. No magsafe light.

These types of problems can be complicated to diagnose as it's both uncommon and possibly caused by any number of issues. Usually this is caused by one or more of the following:

1. Dead battery. The battery doesn't simply have positive and negative terminals, like a typical 9V or AA or whatnot. There's actually a circuit board with a controller built in that controls charging and discharging rates as well as monitors the battery cycles/condition and reports this back to the logic board via the communication pins on the battery connector. A number of issues with this can stop the battery from discharging (ie. powering the logic board) and, similarly, can block charging. One way to check if the battery is having issues is to disconnect the magsafe connector, open the Macbook, disconnect the battery connector, wait a few minutes for any capacitors on the logic board to discharge, then connect the magsafe adapter with the battery out. If it lights up, even briefly, the battery is likely faulty. Also, note that if the Macbook has sat for a long time without the battery able to be charged it is almost certainly 'asleep' and may not accept a charge at this point, requiring a replacement. I'm not sure if this model is able to 'wake' sleeping LiPos but I know some earlier Macbooks weren't.
2. Dead DC-In board. This would prevent the battery from charging as well as (usually) preventing the magsafe adapter from lighting up. It's a fairly easy replacement. If the timespan between when you had it working (when it started working again, you charged it, and then shut down) and when you tried to start it and it wouldn't start was long enough for the battery to completely discharge again a faulty DC-In board is possible. If it wasn't long enough (ie. the battery was fully charged and you shut it down one day and tried to start it the next) then this isn't likely the culprit.
3. Other hardware failure. Failed logic board components or other components could cause these symptoms. The easiest way to test this is to have a donor Macbook of the identical model that's working fine or has a known problem (eg. you bought one on eBay with the screen broken but otherwise fine) you can take measurements or swap hardware to localize the fault. The most likely logic board fault to cause this would be a bad component in the charge control circuit, which should be a relatively minor repair. The expensive components are probably not at fault. Assuming you don't have access to a donor Macbook I'd suggest disconnecting the magsafe adapter and disconnecting components from the logic board one at a time and then reconnecting the magsafe adapter and see what happens. This can't cause any further hardware damage (assuming you practice ESD safety) though your Macbook might not boot correctly. Disconnect the display, the drives, remove memory, disconnect any sub boards (eg. DC In, antennas, etc). After each change reconnect the magsafe adapter and press the power button. Any change in behavior would indicate you've located a likely culprit.

Out of all of these suggestions the battery would be my #1 most likely culprit.