I travel quite a bit internationally.  I make it a rule not to check my bags, so the packing logistics get pretty complex to fit enough clothes for a week long trip into a bag that meets the size requirements for a carry on.  One of the absolute best solutions I have found to getting more changes of clothes into a bag is the use of packing cubes. 

Packing cubes are small zipper bags that come in various sizes and are sold in most travel stores.  Different bags are shaped to optimize for different uses.

I use an Eagle Creek "Half Cube" to pack one day's clothes (minus jacket and shoes).  If you fold your clothes properly, this works great.  I fold my pants in half at the top, then roll them.  For shirts, I fold the shoulders back, then roll from the bottom to the top, careful not to crease the collar.  (Wrinkles can be ironed out in your room, but a creased collar is a bear.)  Undergarments go on top and the whole thing is zipped up, forming a cube.

I've found that in my compact sized carry on, I can fit four or five packing cubes in my bag, or an entire week's worth of clothes!  By removing one carry on, I can usually add a pair of shoes.  Remove two, and I can take a couple jackets.