How does one combine two GCC compiled .o object files into a third .o file?

$ gcc -c  a.c -o a.o
$ gcc -c  b.c -o b.o
$ ??? a.o b.o -o c.o
$ gcc c.o other.o -o executable

If you have access to the source files the -combine GCC flag will merge the source files before compilation:

$ gcc -c -combine a.c b.c -o c.o

However this only works for source files, and GCC does not accept .o files as input for this command.

Normally, linking .o files does not work properly, as you cannot use the output of the linker as input for it. The result is a shared library and is not linked statically into the resulting executable.

$ gcc -shared a.o b.o -o c.o
$ gcc c.o other.o -o executable
$ ./executable
./executable: error while loading shared libraries: c.o: cannot open shared object file: No such file or directory
$ file c.o
c.o: ELF 32-bit LSB shared object, Intel 80386, version 1 (SYSV), dynamically linked, not stripped
$ file a.o
a.o: ELF 32-bit LSB relocatable, Intel 80386, version 1 (SYSV), not stripped