The tricky part is adjusting where the new substrings should start once I found a repeated character. There are 2 possible scenarios:

a b c _a_ b c d

the new begin is string[1], nothing happens to the chars

a b c _b_ a d e f g

the new begin is string[3], and I also have to delete from chars everything up to ‘b’.

def longest_unique_substring(string):
    if len(string) == 0:
        return ''

    chars = [] 
    chars.append(string[0])
    beg = end = 0 # current min/max of string we are processing
    beg_top = end_top = 0 # longest min/max we have found so far
    for x in range(1, len(string)):
        cur_char = string[x]
        if cur_char not in chars:
            # grow the substring
            chars.append(cur_char)
            end = x + 1
            if end - beg > end_top - beg_top:
                # we have a new longest subchar, save it to beg_top/end_top
                end_top = end
                beg_top = beg
        else:
            # found a dupe char - move up until we have no dupes again
            while string[beg] != cur_char:
                chars.remove(string[beg])
                beg += 1
            beg += 1 # don't forget to move this up!
            # beg and end now contain a unique string, continue for loop

    return string[beg_top : end_top]


def test_function(mstr, expected):
    print "Testing %s, expected %s" % (mstr, expected)
    m_res = longest_unique_substring(mstr)
    print " -- got %s" % m_res
    assert(m_res == expected)

test_function('au', 'au')
test_function('abcaccd', 'abc')
test_function('aab', 'ab')
test_function('abcabcd', 'abcd')
test_function('abcaabcd', 'abcd')
test_function('abcabc', 'abc')
test_function('abcbadefgha', 'cbadefgh')
test_function('caabcABca', 'abcAB')