Let's start from the beginning....

A Shimano 105 shifter (for a double ring) should have 3 clicks - one for each ring and one for trim.
A Shimano 105 shifter (for a triple ring) should have 4 clicks - one for each ring and one for trim.

Ok - now, loosen the binder bolt and take the cable off of the FD, so that the shifter is not connected to anything. Move the levers and click it up and down - you should get three clicks both ways. If you do, the shifter is most likely working properly. If not, there is a problem with the shifter.

Next, make sure the shifter is set to the small ring side by actuating the small (inside) lever until it stops clicking. Then, string the cable and check for function. The shifter should be able to click three times up and move the FD with each click. The click is the internal ratchet that keeps the FD from going back down until you use the small lever to release it.

If you cannot get the FD to move up and stay in the last position, then maybe it's not clicking into the final position and you need to adjust the outer limit until it does.

My guess is that you're either not starting in the bottom-most position, or you're not getting into the upper-most position. Either that, or your shifter is broken. Did this shifter work before, or is it a new (to you) replacement?