Bicycle Mechanics - Adjusting Shimano ST-6600 rear derailler

Despite turning the adjusting limiter screw until there is no more motion possible of the idler pulley to the outside, I cannot get the chain to shift into the smallest of the 10 cogs in the rear. The cable is taught, not loose or overly tightened. The gears **** perfectly otherwise. Can't figure it out, but I am new at this.

Inspect shifter cable for kinks. Remove it from shifter and housing. Then pull the entire cable through index finger and thumb. If there is a kink you will feel it.

Adjust the B-tension to set height of upper pulley wheel. Too close and pulley contacts cog teeth. Too low (large gap) causes shifting to hesitate.

Check chain length. If chain is too long the rear dérailleur cage may not be able to keep chain under enough tension to shift.

I'd suggest that you start at the beginning and follow this procedure from start to finish without skipping any steps. If any step does not seem to be going right, find out what the problem is and correct it before going on to the next. http://www.parktool.com/blog/repair-help/rear-derailler-adjustments-derailleur

I would first tug on the exposed cable along the down-tube.

Does the derailer "clunk" when you release the cable suddenly?

If there is no clunk, then the cable is either too tight or the cable housing or bb cable guide (bottom of frame) are clogged with dirt/sludge.

At minimum, I suspect you will need a new length of cable housing going into the rear derailer, but it is usually possible to un-hook the cable housing from the cable housing stop on the chainstay and then slide the housing up and down the cable so as to get it freed up, cleaned and lubed so the cable slides more smoothly.
To get the housing un-hooked, first shift the chain to the biggest rear sprocket, then shift the shifter to the small sprocket position WITHOUT TURNING THE WHEEL OR CRANKS, which will totally slacken the cable and allow the housing to be un-hooked.