No major dictionary that I'm aware of lists "eachother" as a word, but I don't understand why. It's a common construction, means something different than "each other", and in common speech it generally gets pronounced differently than "each other" as well, with the two words blurring into eachother.
"They looked at each other block."
"They looked at each other."
"They looked at eachother."
To my eye, the first and third make syntactic sense, while the second looks like a sentence fragment.
So why isn't it acceptable as a single word? By all the logic I can see, it functions as one already. I realize this is more of a rant than a question, but I would welcome some grammarian feedback on this issue if there's any to give.