Shouldn't the small AC input current in a common emitter amplifier be amplified to DC?
What you are missing is the concept of bias point.  Note Vbb in your schematic.  That is adding (presumably) just the right DC offset so that when no additional signal is applied, the collector voltage is roughly in the middle of its range.  The AC input signal then adds and subtracts from the center point.  If the AC input signal gets too large, then one or the other limits will be hit and you do have clipping.

By the way, this open-loop method of biasing a transistor is OK for theoretical analisys where you can just assume the bias point is correct.  However, in practise it is pretty much impossible to find a fixed value of Vbb to hold the circuit near its midpoint of linear operation.  This is usually done by feeding back some of the collector voltage to derive the bias level, or by adding a emitter resistor (which is in itself a form of feedback too).

Also, please use the right symbols in your schematics.  Look up how a NPN transistor is supposed to be drawn.  There is more on drawing schematics at https://electronics.stackexchange.com/a/28255/4512.