Rear shifting - Unable to shift to toughest gear I was adjusting my rear shifting following the advice in this video: 
                
            

It was when I took off the wire tension and screwed the high limit all the way out that the chain wouldn't make it to the toughest gear. Instead it was stuck at the second-toughest gear.

So it seems that even without the wire pulling on the derailleur, it is too far on the left (i.e. towards the easier gears). What could I try to track down the problem, or better, what exactly is causing this and how can I fix this?

I was then continuing to adjust the derailleur, making the chain switch to my second-toughest gear when I try to shift to the toughest gear with the shifter at my handlebars and making the chain switch to my easiest gear when I try to shift to my second-easiest gear. This worked fine and the chain is currently not skipping gears on its way up or down, it just can't reach the toughest gear.
The first thing to do with any rear derailleur problem is to check if its bent. If it is bent, go to your LBS and see if it can be unbent (there is a special tool for checking unbent-ness, Park Tool DAG-2 or similar) -- bent derailleurs cannot be adjusted properly in most cases (and even a minor spill can cause a bent derailleur). If its not bent, then try adjusting it. 

If you can't reach the smallest cog with the cable released (assuming a typical non-rapid rise derailleur, i.e. one which goes to the smallest with no cable tension), it likely means your high limit screw is mis-adjusted (too tight). [Typically, the problem upshifting into the smallest cog is usually a bent derailleur/cable friction, but you should have eliminated those two options before you got to this]. 

Some good tutorials for adjusting derailleurs are Sheldon Brown and Park tool.