There are several approaches to 'add' more polygons
You can open the menu 'specials' by pressing W while beeing in edit-mode, where you can find 'Subdivide' and 'Subdivide Smooth'.
Note: In the Toolbar (toggle with T) options will be displayed.
You can press Ctrl + R to cut edges. This will add more polygons. https://www.blender.org/manual/modeling/meshes/editing/subdividing/loop_subdivide.html
By pressing shift + A you can open up the mesh-menu, where you can select and insert a mesh.
By pressing E you can extrude your selected vertices, edges or faces.
There are multiple extrude-modes that you can find pressing space and tiping 'extrude'. That will display all actions with 'extrude' in their name.
By pressing I Takes the currently selected faces and creates an inset of them, with adjustable thickness and depth. If several faces are selected you can choose to inset them individually.
By pressing K you can open the 'Knife Tool' where you can cut faces, what will create more polygons.
Although this can't be made in edit-mode. In Object mode select multiple meshes and join them pressing Ctrl + J