Foreign tongues are a fascination of mine. When I was in high school I embarked upon teaching myself Japanese, an effort which continues to this day (and this was at a time when appropriate books were hard to find--you punks don't know how good you have it these days). I remember being struck by the assumption in one of my books that one must be planning a trip to Japan in order to be studying Japanese. At the time I had no such prospects. Growing up where and when I did, I had never even seen a real Japanese person.
Why study, then? It is logical to study in preparation for travel--but in my case it was more as a substitute for travel. By studying the language of a foreign country, you immediately get inside the heads of the people there. In some ways, you are already more intimately acquainted with the place than someone who goes there physically but has no understanding of the language. For this reason it has always been difficult for me to resist the call of a mysterious new language. I have a particular weakness for those written in exotic, alien scripts.
A special category consists of the dead languages, for these carry you back in time as well as across space. Study Latin and visit Imperial Rome, for example. Languages such as Latin and Sanskrit never died out completely (and were widely used even after all the "native speakers" had disappeared), but for several ancient scripts--Egyptian hieroglyphs, Linear B, etc.--the knowledge of how to read was lost altogether.
In several cases tiny clues have been leveraged with great insight to recover the ability to read the script. In cases such as Mayan this has opened up whole tomes of forgotten history. Such a decipherment has never been accomplished without some kind of clue--a bilingual inscription such as the Rosetta Stone for Egyptian, or a resemblance to a known language, such as Greek for Linear B.
If you want to try your mind against a dead language, I suggest you try Sumerian (pic above), which once was spoken in parts of what is today Iraq. It's the ultimate dead language. Here's why.
Cuneiform has been used across Asia Minor to what is now Iraq and Iran. Cuneiform is not a language, nor even a single script, but a method of writing with a wedge-shaped stylus on tablets of clay (or occasionally more permanent media such as stone or metal). You can imagine how unwieldy a clay tablet is compared to a sheet of paper, but for us it works out better because clay is so much more durable. In fact, a most fortunate event from our standpoint was for the library to burn down, because the clay tablets would be baked to a permanent hardness.
Among the peoples who used cuneiform, the Assyrians (a.k.a. Babylonians) were major players. They spoke a language related to Arabic and Hebrew. This was one of those languages whose knowledge had died out, but it was deciphered in the 1850's. The connection with Arabic and Hebrew was key. And then....
Among the various Assyrian tablets deciphered were found word lists--dictionaries, in other words--and instruction books used by the Assyrians to study yet another language, which to them was an ancient language, whose very existence had been forgotten by the 1800's. This was Sumerian. And working at two levels removed, the millenia-old word lists compiled by the Assyrians were the key to deciphering Sumerian, which seems to be unrelated to Assyrian or to any other known language, ancient or modern.
Sumerian is the dead language's dead language. You must admit, that's pretty cool.