What you are describing is call a blind via. With current technology, they are limited to 1:1 aspect ratio because of plating difficulty in a blind hole. The 1:1 ratio severly limits the practical depth of a blind via.
Mechanically drilled blind vias are not generally done because of the difficulty in controlling the depth, difficulty in cleaning the hole for plating, and the fragile nature of the fine drills that would be required. The smallest drill size that a mechanically drilled blind via can practically use is about 20mils, which results in a finished size of about 15mils (which also means a max depth of 15mils). If you want a smaller via, then laser drilling is the only alternative.
Yes... blind vias will cost you much more than thru-hole vias. The reason for the extra cost is the greater number of steps added to the board production, and the decreased yield in finished boards.
See the following for a description: **broken link removed**