What options to grow object? I tried to animate my object to appearing like from nowhere. I tried to use boolean's intersect modifier so that I had a box over my object which I animated from down to up.
Very much the same what Chris does on his example:
Sci-Fi Laser 3D Printer Effect
My issue was that sometimes my object totally vanished while the box was coming up and showing more about my object. Or when looking for this from top, there were no faces on up, like he had at 3:19 on his video. I believe the problem could be my object itself. I tried to fix that by merging the closest vertexes, but did not end to perfect solution.
I also tried to use build modifier, but that I feel was a bit uncontrolled, and really did not give the effect of &quot;extrude&quot;.
Is there some other options which I did not thought to get similar effect? Or shall I try to continue to fixing my object to get this work?
You could use the Mask modifier.
First you need to paint some weights to vertices in Weight Paint mode.
Go to Weight Paint mode and select your object. Add a vertex group and select Gradient tool. Pull the tool across your object and it will paint the weights as can be seen in the image below.

Next go to Object Mode and give your object a Mask Modifier. Pick the Vertex Group you created earlier and adjust the Threshold. Threshold can be keyframed to make the object appear and disappear.

The more subdivisions your object has the smoother the appearing is.