I’d use Enum.reduce_while:

result = Enum.reduce_while(String.graphemes("ABCDAHWX"), [], fn character, prev ->
  if character in prev, do: {:halt, character}, else: {:cont, [character | prev]}
end)

with char when is_binary(char) <- result do
  char
else
  [_ | _] -> :no_duplicates
end

https://www.regular-expressions.info/

https://regexr.com/ set the RegEx Engine to PCRE (Perl Compatible Regular Expressions)

Of course, I expect the regular expression version to be faster

iex(2)> :timer.tc fn -> Enum.each 1..1_000, fn _ -> Test.first_recurring("ABCDAHWX") end end
{1876, :ok}
iex(4)> reg = ~r/(.).*?\1/
~r/(.).*?\1/
iex(6)> :timer.tc fn -> Enum.each 1..1_000, fn _ -> Regex.run(reg, "ABCDAHWX") end end
{5585, :ok}
iex(7)> :timer.tc fn -> Enum.each 1..1_000, fn _ -> Regex.run(reg, "ABCDAHWX") end end
{3823, :ok}
iex(8)> :timer.tc fn -> Enum.each 1..1_000, fn _ -> Regex.run(reg, "ABCDAHWX") end end
{4058, :ok}

Although the regex pattern is not compiled, but the elixir version (above) is not optimized either. I usually find regexes to be much slower than erlang’s pattern matching since they do much more work.