Not sure about how many containers, but several hundred of them:
Error: Cannot destroy container 0c288ee3edb3: Driver aufs failed to remove root filesystem 0c288ee3edb3bae7829be63ed84c460ca3b5d0a480b1f780f31a12e083957ad2: open /proc/self/mountinfo: too many open files
Not sure about how many containers, but several hundred of them: