Which unit cell do we obtain if spheres are arranged in ABABAB type in 2D and then that 2D layer is arranged in AAAAAA type in 3D? Arranging spheres ABABAB type in 2D gives us close packing. Now, this layer if arranged in ABABAB (in 3D) gives us HCP and if arranged in ABCABC (in 3D) gives us FCC. 

What do we get if we arrange it in AAAAA type (in 3D)? (we are talking about one type of spheres only)
The question is a little confusing, so I will interpret your 'ABABAB' type in 2D to mean a simple hexagonal layer (so, the same layer as used for your hcp and fcc packing). So, a stacking of a simple hexagonal layer, each layer directly above the other, will give a structure looking like right hexagonal prisms - or right triangular prisms if you look a bit more carefully.  

Anyway, the resulting structure is the simple hexagonal Bravais lattice with a single atom basis. Hcp is also a simple hexagonal Bravais lattice, but with a two atom basis, one each from plane A and plane B. I do not know of any elements that crystallize in this crystal structure (but am open to corrections!).