Not using regexps at all, but...

#!/usr/bin/env perl

use 5.012;
use Test::More;

sub find_substring
{
    my $input  = shift;
    my $length = length $input;
    
    for my $i (1 .. $length)
    {
        my $possible = substr($input, 0, $i);
        my $repeated = $possible x (1 + int($length / $i));
        return $possible if $input eq substr($repeated, 0, $length);
    }
    
    return "";
}

my %eg = (
    abcdabcdabcdabcdab          => "abcd",
    abcdabcdabceabcdabcdabceab  => "abcdabcdabce",
    aaaabaaaabaaaaabaaaab       => "aaaabaaaaba",
);

for my $input (sort keys %eg)
{
    my $expected = $eg{$input};
    my $got      = find_substring($input);
    
    is($got, $expected, "result is '$expected' given input '$input'");
}

done_testing;
[download]

Note that when there are multiple possible matches, this returns the shortest, because it doesn't make sense to return the longest - the longest is uninteresting.

For example, given the input abcabca, it could be that the answer is abc repeated two and a bit times, or abcabc repeated one and a bit times, or abcabca repeated exactly once. (Well, not really "repeated" but you know what I mean. The entire input string itself is always a valid and uninteresting answer.) Or, depending on how the problem is defined, the correct answer might be abcabcaxx repeated less than one time - i.e. the first repetition was truncated!

So the only interesting answer to return is the shortest possible one.

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