How do you view your unranked ELO rating?

A couple posts I've found on the LoL forums suggest looking in some "air" directory under your profile's logs, but I have no idea where this is. How can I find my ELO ratings in League of Legends?

Unranked elo used to be accessible via a glitch, but isn't anymore. However, it still exists, and is used for matchmaking. (it just isn't visible)

Ranked elo is accessible on profile and visible by other players if you have an elo of 1250 or more.

You are not able to explictly view your Elo. This is by design in order to keep unranked matches casual. However, you can get an idea what your Elo might be, including the Lolcave tool mentioned by Sadly Not.

Another way to get a rough estimate is to peek at the ranked Elos of the players you get matched with.

Finally, here are some hints to give you an idea:

• Do teams always have a jungler? You are high Elo. Do teams rarely have a jungler? You are low Elo.
• How long is the laning phase? Over 20 minutes and you are probably low Elo.
• Is Dragon attacked (or attempted) whenever it's up? You are high Elo. Is dragon usually ignored? You are low Elo.
• Does everyone buy wards and is the map usually fully warded? You are high Elo. Do players rarely buy wards? You are low Elo.
• Are buffs usually ignored? You are very low Elo.
• Do players help their teammates get minion buffs? You are mid to high Elo.
• Are players last-hitting minions? You are mid to high Elo.

Edit

As of the Pulsefire Ezreal patch, Riot changed the protocol and ELO rating is no longer sniffable by third party tools (source). My understanding is that they want unranked games to be a less competitive mode than ranked. There is no need for ELO values to ever leave the server, since matchmaking happens on the server side. That leads me to believe that they will never publish unranked ELO, either directly or indirectly, again. I'll leave my original answer below, but it does not reflect the current state of the game.

Some third-party apps can tell you your unranked ELO. I know that LolMatches does. Essentially, they sniff the interactions between your game and the server, and are able to parse out your ELO. That means that the numbers they get are 100% accurate, and not a calculated estimate based on fuzzy win/loss stats.

As an example, you can take a look at my profile. The search button is a little hard to find; it's in the top left, near the logo. You might already be in the database if you've played a game with another player running the client.