The console.log will run before the ajax call is ended; write your code or you console.log in your success function right after the end of the data loop.
Like:
$liked = 'https://api.instagram.com/v1/users/self/media/liked?access_token=' + $access_token + '&COUNT=20';
$images = []
$.ajax({
type:'get',
dataType:'jsonp',
cache: false,
url: $liked,
success: function(data) {
for (var i = 0; i < data.data.length; i++) {
console.log(data.data[i].images.standard_resolution.url);
$images.push(data.data[i].images.standard_resolution.url);
}
console.log($images.length); // here the array is filled with response data
}
});