CDO

Max of each Field:

cdo output -fldmax infile

cdo output -fldrange infile

Thank you, I thought about fldrange, but didn't understand the syntax.
And if I try to get this value for one field within an extent of another one, would the "-remap,infile1 infile2 out -> output fldrange,out" be the only and best way?

You can combine these commands to one line:

cdo output -fldrange -remap,... infile > out

Hm... And why
cdo -setname,var777 -mul -div -output -fldrange -select,name=var199 $infile1 -output -fldrange -select,name=var182 -remapnn,$infile1 $infile2 -select,name=var199 $infile1 $outfile
does not work?
We cannot parallelize outputs? It should be a field '-select,name=var199 $infile1' multiplied by a calculated constant '-div -output -fldrange -select,name=var199 $infile1 -output -fldrange -select,name=var182 -remapnn,$infile1 $infile2'

The CDO operator output can't be combined with other operators. It simple print the contents of a data stream to screen.
You have to remove this operator:

cdo -setname,var777 -mul -div -fldrange -select,name=var199 $infile1 -fldrange -select,name=var182 -remapnn,$infile1 $infile2 -select,name=var199 $infile1 $outfile

This does not work, because it says: cdo(2) mul (Abort): Grid size of the input parameters do not match!
Nor mulc operator helps.

Ok, the field size of the input streams for mul need to be the same. You can use enlarge for this task. I have also replaced select,name by selname:

cdo -setname,var777 -mul -enlarge,$infile1 -div -fldrange -selname,var199 $infile1 -fldrange -selname,var182 -remapnn,$infile1 $infile2 -selname,var199 $infile1 $outfile