On my PC (which is a long way from being a “Supercomputer”) this is what I get on a fresh kernel:

In[1]:= 
MemoryInUse[]
MaxMemoryUsed[]
Out[1]= 97675416
Out[2]= 98118248

In[3]:=
F[x_,y_,g_]:=x-(Exp[y]-Exp[-g*y]);
y0=2*ArcSinh[x];
x=10;
g=10^10;
y/.FindRoot[F[x,y,g],{y,y0},WorkingPrecision->50,PrecisionGoal->30,AccuracyGoal->\[Infinity]]
Out[7]= 2.3025850929940456840179914546843642076011014886288

In[8]:= 
MemoryInUse[]
MaxMemoryUsed[]
Out[8]= 94880360
Out[9]= 10908680648

Finding the root takes less than 5 seconds

My PC is a pretty basic Asus X415JA. Installed RAM is 8 GB. SystemInformation gives:

The virtual memory may contribute to explaining that 10 GB can be handled.

Just to confirm: Mariusz's code works without trouble on "14.1.0 for Mac OS X ARM (64-bit) (July 16, 2024)"

It seems to be a Windows problem. (Maybe Linux?)

I run the test code in batch mode under version 14.1 on a supercomputing cluster, and there is an insufficient memory error independently of whether I request 3850 MB or 7700 MB of memory. I have not tried more, because this seems nonsensical to me to expect that so much memory is needed. In the batch mode, I believe, everything is cleared before any program is run, as every run begins with loading the MATHEMATICA package. So, I still don't know what might cause such a memory demand. I would appreciate any suggestions. Some other of my tests suggest that changing the method to Brent makes things even worse. Is there here any potential problem with overflow or underflow in Exp? Maybe Newton iterations cause the consecutive approximations to the root to escape into regions of overflow or underflow? Is there any way to get an information about how these consecutive approximations look like?

Leslaw