I have built a couple of two element wire Yagis. They worked pretty good. I got a certain 3 dB improvement possibly 5 at the most. The two element Yagi antennas are almost fool proof. They can work as unidirectional with pretty good front to back or as bi directional with pretty good gain in both directions. I had one that had the bi directional feature. It may sound strange at first but you can build a two element Yagi and depending on the spacing and length of the parasitic element it can be either a director or reflector and somewhere in the middle it acts as both!  I attached a chart below from an antenna book. The driven element and parasite in this special case are the same length.  As the spacing is increased to 0.1 wavelengths it develops good gain, a little over 5 dB,  as a director in the A direction. As the spacing is increased more it loses some gain in the A direction and develops gain in the B direction. At 0.15 wavelength spacing it has about  equal gain (4dB ) both directions. If the spacing is increased further the gain in the B direction levels out at about 5 dB while the gain in the A direction drops.

With the parasite as a director you never get great front to back but as a reflector and spacing of 0.25 to 0.3 wavelength you get pretty good front to back.  Of course there is an infinite number of combinations when you start making the parasitic element longer and shorter than resonance.

It’s hard to build a two element Yagi that does not work. You will get gain in one or both directions even if you don’t measure right! 

The general rule of thumb is that in order to get any noticeable gain you have to double the antenna size, both number of elements and boom length. So two elements gives you the most bang for your buck over a single dipole. The next point is 4 elements and double the boom although going to 3 elements might still be worth it, that’s debatable. Then to see a difference you go to 8 elements and again double the boom length. Adding just one element like going to 5 after you already have 4 is insignificant. The law of diminishing returns is at play. Also at some point the coupling between parasitic elements decreases just due to the length of the boom. That’s one reason you see stacked Yagis like 4 over 4. That gives increased gain ( like 3 dB) and also usually eliminates a null in the elevation angle. The big improvement is not raw gain but the absence of a deep null.