I know you said "without third-party tools", but sometimes a tool is just too obviously the right one, plus it's installed on most Linux systems by default:
For those without rev (recommended), there is the following simple awk solution that splits fields on the null string (every character is a separate field) and prints in reverse:
The above awk code is POSIX compliant. As a compliant awk implementation is guaranteed to be on every POSIX compliant OS, the solution should thus not be thought of as "3rd-party." This code will likely be more concise and understandable than a pure POSIX sh (or bash) solution.
(; I do not know if you consider the null string to -F a regex... ;)
Nobody appears to have posted a sed solution, so here's one that works in non-GNU sed (so I wouldn't consider it "3rd party"). It does capture single characters using the regex ., but that's the only regex.
In two stages:
$ echo 123456 | sed $'s/./&\\\n/g' | sed -ne $'x;H;${x;s/\\n//g;p;}'
654321
This uses bash format substitution to include newlines in the scripts (since the question is tagged bash). It works by first separating the input string into one line per character, and then by inserting each character into the beginning of the hold buffer.
x swaps the hold space and the pattern space, and
H H appends the (current) pattern space to the hold space.
So for every character, we place that character into the hold space, then append the old hold space to it, thus reversing the input. The final command removes the newlines in order to reconstruct the original string.
This should work for any single string, but it will concatenate multi-line input into a single output string.