Well, I think the col -b solution was the best solution, but one style of regex for your purpose could be:

s/([a-zA-Z])\1/$1/g; #happy now
[download]

However this will clobber lines like this one. Also we are not considering non-alpha at all.
Update: And eg has the perl solution that will work. (Could of sworn that's what I said in the CB)
YAU: I'm assuming the downvote came from my use of \1 instead of of $1. (Lots of vi editing always does that to me.) If not, please /msg me and enlighten me.

For those that are wondering about the \1 & $1: Inside a match use the \1 backreference. Outside use the $1 notation. This doesn't wind up being very important unless you've got more than nine backreferences. \10 is shorthand for \010 which is octal. So if you have 10 or more matches \10 will be the tenth match, otherwise it's octal 10.