Occasionaly I run across this sort of thing in printed text:
It just so happens that that is the way…
See what I mean? The same word juxtaposed to itself within a sentence. While it doesn’t cause my heart to stop beating, it often encourages a double-take while reading. The same occurs with other words as well, though I can’t put my finger on another example at the moment. So is it proper grammar? I’m thinkin’ it ain’t.
Yes, it is technically correct, though good writers will reword the sentence to avoid it. The first “that” functions as a conjunction, while the second “that” functions as a pronoun. Grammatically correct, but awkward nontheless.
I think you’re right about there being other words this is possible with. I’ll see if I can come up with one.
Carpe hoc!
Not that that’s wrong.
MrKnowItAll:
Answers my question and then ends a sentence with a preposition? How about MrKnowsQuiteABitButNotQuiteEverything. 
An infuriating British person I worked with once asked me if I could think of a way in which the word “and” could be used 5 times in a row. The answer is: say for instance you’re passing a pub called the Pig and Bear (or something) and the words look too close together. You would say “There should be more space between ‘Pig’ and ‘and’, and ‘and’ and ‘Bear.’” You would? I wouldn’t! But technically it’s right, I suppose.
I had had some deep thoughts about this very subject recently, but I forgot them.
President of the Vernon Dent fan club.
Gee, sly, you should talk to Rowan.
I thought that whole “preposition at the end of a sentence” thing had finally been tossed in the dumpster where it belongs.
Carpe hoc!
Do you know where I can find a dumpster to throw that rule in?
My favorite example along these lines was from Steven Pinker’s “The Language Instinct,” in which he points out that, assuming there is a variety of buffalo from the city of Buffalo, the sentence “Buffalo buffalo buffalo Buffalo buffalo” is perfectly parseable.
And there’s the old Latin gag: Malo malo malo malo, which roughly translates: I’d rather be up an apple tree than a bad man in adversity."
John W. Kennedy
“Compact is becoming contract; man only earns and pays.”
– Charles Williams
There is a grammatically correct way of saying something that nevertheless gets snickers if you do do it.
I believe the first “that” could be left out. I read an article in a syndicated language column on this subject a few years ago that said “that” as a conjunction can be left out. For instance, it is correct to say either:
It just so happens that I am an expert.
It just so happens I am an expert.
The second sentence is even preferable because it saves just that much more time and space in printing, typing, speaking, what have you. At work, when I compose a document, I try to leave out that “that” to show how clever I am, but our managing staff attorney always puts them back in 
“I hope life isn’t a big joke, because I don’t get it,” Jack Handy
Puctuate the following sentence:
Jim where Bill had had had had had had had had had was right.
Answer:
Jim, where Bill had had “had had”, had had “had”; “had had” was right.
I don’t get the one about the Buffalo buffaloes. Are they buffaloing themselves?