Q:

Can a low-wattage adapter fully recharge the battery of a turned off MacBook? 

MacBooks come with power adapters that provide a certain wattage (for example 87 W) required to keep the computer running and charge the battery.  If used with a lower-wattage power adapter (for example 61 W), there might not be enough power to keep the computer running.  However, could such a lower-wattage power adapter be used to fully recharge the battery if the computer is kept turned off? 
Could even a low-wattage iPhone charger be used for this, although it would probably take a long time? 
Update: The question concerns newer MacBooks with USB-C chargers. 

A:

Assuming you are referencing the MacBooks and MacBook Pros with USB-C ports (2016 and up), the short answer is yes, a low power adapter could charge the MacBook which requires a higher power adapter to function. 
(and yes, it will take a very, very long time)
As for Macs with the MagSafe adapter, since the DC-In board (or MagSafe power board) "checks" the power coming it, if it's not of the right voltage/wattage, it won't charge. 

A:

Yes, it will charge from a lower-wattage adapter so long as the computer is not burning the energy faster than the charger is providing it. 
I personally have a 15" 2016 MBP.  It came with an 87W adapter, which I usually leave at work.  At home, I have an Anker USB-PD 60W adapter.  Typically lack of wattage isn't noticeable, but on a few occasions, while running the MacBook Pro hard, I have noticed the battery is not charging, or even draining.  In that case I need to calm things down or plug in the 87W adapter.* 
Anything with a USB-PD charger should be able to charge the battery, even the 30W model, though in that case your MacBook Pro may need to be asleep to charge. 
*Fun fact, you don't actually need to unplug the 60W adapter in this case, the computer will draw from the higher wattage adapter, and stop drawing from the lower.  See this question. 
Update in 2020
I've now charged on low-power chargers countless times.  I've even recharged overnight with a 3A phone charger when I didn't have anything better.  It takes hours but you'll have a full charge in the morning. 
Theoretically, the 15" Macbook's 7.2Ah battery would charge in 4 hours assuming 60% efficiency, that the charger actually puts out a full 3 amps, and there is no usage during that time.  At 2.4A, you'd be around 5 hours and so forth.