No.
As others have said, DOS by itself is single-task, single-application.
Individual applications can implement a "call out to DOS" functionality, (including DOS Shell, included with MS-DOS 5.0 or higher.) But the DOS kernel / Command.com itself has no such functionality. When using the "call out to DOS" functionality, you are at the mercy of that application's implementation of swapping itself out and running Command.com in the remaining memory.
So in your example, when you boot to MS-DOS, you are running the single task "command.com"
Then when you run Links, command.com removes most of itself from memory, and loads links.exe with the instruction "when you exit, reload this memory address" which contains the bit of command.com that stayed in memory.
When you invoke Links' "drop to DOS" function, Links removes most of itself from memory, and loads ANOTHER command.com in to memory, with the instruction that "when you exit, reload this memory address" which contains the bit of Links that stayed in memory. Note that because the bit of memory the first copy of command.com left behind is in use, Links instructs to use a different memory address.
From that DOS, you load Word Perfect. The little bit of Links is still in memory, the little bit of the original command.com is still in memory, and this time around, the "new" command.com leaves a little bit of ITSELF in memory, too, telling Word Perfect "when you quit, reload this memory address"...
Each time, you are launching a new process, and the old "reload this memory address" pointers are not *ACTIVE* at all. They are simply instructions left alone in memory, waiting to be called. If you never quit Word Perfect, those addresses are never called, neither copy of command.com is ever actively doing anything, and Links never actively does anything.
Again, if you drop to DOS from Word Perfect, you're just repeating the cycle. Word Perfect unloads most of itself, leaving behind only a little bit, with the instruction to the new (THIRD!) command.com to call that address when it quits. That new third command.com only knows about Word Perfect's "callback address", it is completely unaware of the bits of the other two command.coms, or Links. Oh, it knows that those bits of memory are unavailable for its use, but it has no directions to call them for any reason.
When you type "exit" at a command.com prompt, it really just tells command.com "hey, call back to that address you were told about when you loaded and run that." The DOS from Word Perfect only knows about the address to reactive Word Perfect. Word Perfect (when you quit) only knows about the address to reactivate the command.com that Links launched. The command.com that Links launched only knows about the address to reactivate Links. Links only knows about the address to reactivate the initially-booted command.com. The initially-booted command.com knows that there is no "callback address," so typing exit there does nothing.
Note that if you look at the autoexec.bat created by Windows 95/98's "Boot to DOS" mode, it actually "calls" command.com, with the exit command of "win.com" - so that this DOS mode understands "when you exit, run win.com". (Which then loads enough to realize it had been launched that way, and reboots in to "normal Windows 95/98".)