Far too many Strings. Why don't you store just a single character and count its repetitions, then copy the char and the number to some maxchar and maxnum if number > maxnum?
EDIT (code added):
public static int maxRun(String testString) {
int len = testString.length();
char maxchar = "!";
int maxlen = 0;
char prevchar = "!";
int sublen = 0;
for (int i = 0; i < len; ++i) {
char curr = testString.charAt(i);
if (curr == prevchar) {
++sublen;
}
else
{
if (sublen > maxlen) {
maxchar = prevchar;
maxlen = sublen;
}
prevchar = curr;
sublen = 1;
}
}
if (sublen > maxlen) {
maxchar = prevchar;
maxlen = sublen;
}
for (int i = maxlen; --i >= 0;)
System.out.printlnprint(maxchar);
System.out.println("");
return maxlen;
}