Any more unstated rules? :)

C:\test>876075
AGRTGAXWXX : [ AGRTGA ]
ACRMGAHKMAHGTXX : [ ACRMGAHKMA, GAHKMAHGT ]
AXXAXXAXXXXXXXXXXXXXXAXXA : [ AXXAXXA, AXXA ]
[download]

#! perl -slw
use strict;
use Data::Dump qw[ pp ];

sub maxMatches {
    my $s = shift;
    my @matches;
    my $vec = '';

    for my $o ( 0 .. length( $s ) - 10 ) {
        my( $match ) =  $s =~ m[.{$o}([ACGT].{0,8}[ACGT])] or next;
        my $mask = '';
        vec( $mask , $_, 1 ) = 1 for $-[1] .. $+[1]-1;
        next if ( $vec | $mask ) eq $vec;
        $vec |= $mask;
        push @matches, $match;
    }
    return @matches;
}

while( <DATA> ) {
    chomp;
    printf "$_ : [ %s ]\n", join ', ', maxMatches( $_ );
}

__DATA__
AGRTGAXWXX
ACRMGAHKMAHGTXX
AXXAXXAXXXXXXXXXXXXXXAXXA
[download]