Find out how many times the user has logged in

I don’t think so. You will need to do something like this (untested):

add_action("wp_login", "my_login_function");

function my_login_function($username){
   $userdata = get_user_by('login', $username);
   $n = get_user_meta($userdata->ID, "my_login_counter", true);
   if (! is_numeric($n)) $n = 0;
   $n = intval($n) + 1;
   update_user_meta($userdata->ID, "my_login_counter", $n);
}