First, I should let you know that I've just started homebrewing. I received a 1 gallon DIY kit as a gift, and after three batches I made the jump to 5 gallon batches and a kegging system. I'm trying to learn everything I can about homebrewing, and fortunetly I'm making a lot of mistakes (opportunties for learning) along the way.
On my very first 5 gallon batch (after fermentation), I accidentally pushed the rubber stopper too far into the secondary carboy. Since it was still airtight, i just left it in there, and when it came time to transfer to the keg, I pushed the rubber stopper into the brew and completed the transfer. A quick Google search brought up a lot of questions on how to remove it, but I couldn't find anything substantial about actually removing it.
So I looked at what tools I had available and tried to figure out my own solution. First, you will need one of those utility line flag markers. Bend the metal end into a U shape, long enough to hook the rubber stopper, and still have enough length on the other end so that both stick outside of the carboy after you've hooked the stopper.
To hook the stopper, turn the carboy upside down so that it sits in the hole in the opposite direction that you put it in (narrow side to the hole). Insert the metal utility line flag and pull down so that the other end of it goes threw the hole. Hold the stopper in place and place the carboy on a table, right side up.
Now you should be able to hold down on the carboy and pull up on the metal wire. I found that bending it at the lip of the carboy and using the neck as leverage was easiest when pulling up the stopper. If you have a good pair of needlenose plyers, you can try to grab both ends of the metal wire and pull the stopper out. When the stopper was halfway out of the neck, it got harder to pull out, i assume this is due to friction, so i poured a little water on top of it and continued to pull. finnaly the stopper poped out of the hole.
The actual hooking and pulling took less than a minute. I've learned a lot on this website and hope that this can, in turn, help you.
Utility line marker with hooked stopper
Success! The stopper has been extracted!