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UPDATE: I have added a "Contributions" sections at the end this entry to showcase my personal favorite TRFs suggested by other Deviants. Naturally, the original contributor will always be credited.
Teddy Roosevelt wasn't awarded the Medal of Honor until 2001, because it took 82 years to fully comprehend his badassery.
Teddy Roosevelt started a cattle ranch because he refused to eat anything he hadn't killed with his bare hands.
Teddy Roosevelt quit his job as a New York City Police Commissioner after only 2 years, because he had run out of criminals to punch.
Teddy Roosevelt was appointed Assistant Secretary of the Navy after he punted a Battleship from the Gulf of Mexico into Lake Superior.
Teddy Roosevelt could beat Rock with Scissors.
Teddy Roosevelt's version of a "Slam Dunk" did not involve jumping and scoring, but crushing and maiming.
Shaking hands with Teddy Roosevelt would have been considered sufficient grounds for an insanity plea in a court of law, if there had ever been survivors.
President McKinley did not actually die from a gunshot wound, but from overexposure to Teddy Roosevelt's badassery. He had no regrets.
The Megalodon shark was 67 feet long, weighed 65 tons, and was the deadliest predator to ever live. Teddy Roosevelt used them as bait.
Merriam-Webster's Unabridged Dictionary of the English Language defines "Badass" as "Teddy Roosevelt". The same holds true for the words "Deadly", "God-like", "Invincible", and "Apocalyptic".
Teddy Roosevelt could swallow nails and chew glass, but he didn't like eating "sissy food".
Teddy Roosevelt ended the Russo-Japanese War by flexing.
Teddy Roosevelt refused to enter any swimming pool that hadn't been stocked with starving piranha.
Teddy Roosevelt's glasses didn't help him see better, they just shielded others from his Death Gaze.
A Passenger Pigeon once crapped on Teddy Roosevelt's new hat. There are no more Passenger Pigeons.
Originally, Mount Rushmore was going to be four carvings of only Teddy Roosevelt, but it was determined that having so much badassery concentrated in one place might destroy the universe.
Teddy Roosevelt introduced the yard and the foot as units of measurement; they were determined by the length and width of his manhood, respectively.
Teddy Roosevelt personally dug the last mile of the Panama Canal, unassisted. He used a single punch.
Teddy Roosevelt only decided to die because he had already killed everything worth killing. He's in the process of re-killing it all in the afterlife.
Repeats
Teddy Roosevelt carries a big stick to give his prey a sporting chance; his fists would be too deadly.
Teddy Roosevelt became leader of the Bull Moose party after killing a bull and a moose with his bare hands. In his sleep.
Teddy Roosevelt refused to kill a trapped bear cub because it was his nephew.
The Rough Riders were, in fact, all Teddy Roosevelt.
When Teddy Roosevelt was shot, the bullet knew better than to try to actually wound him.
Someone made fun of Teddy Roosevelt's glasses once. Once.
The only thing more deadly than Teddy Roosevelt, is Teddy Roosevelt pissed off that someone implied there was anything deadlier than him.
A barber once tried shaving Teddy Roosevelt's mustache. There were no survivors.
Teddy Roosevelt quit his hobby of being President because it was interfering with his career of killing deadly animals with his bare hands.
Every dinosaur skeleton on display in a museum is a hunting trophy donated by Teddy Roosevelt.
Chuck Norris grew a beard to go with his mustache because he didn't feel worthy of copying Teddy Roosevelt.
Pluto used to be Earth's closest neighbor, and larger than Jupiter. Then it pissed off Teddy Roosevelt.
Contributed
Abraham Lincoln: Vampire Hunter was actually based on Teddy Roosevelt.
Teddy Roosevelt created the Grand Canyon by suplexing a freight train.
It is widely known that nobody can defeat Chuck Norris; what is less widely known is that Teddy Roosevelt's middle name is "Nobody".
Teddy Roosevelt wore pince-nez glasses to prove that he could make anything look badass. He succeeded.
Teddy Roosevelt founded the founding fathers.
Most people consider an enraged African Elephant to be a living natural disaster. Teddy Roosevelt considers them target practice.
When Teddy Roosevelt wrote a book, he never had to use a pen or typewriter. He simply ordered the words to appear on paper, and they obeyed out of sheer terror. 
Zeal 
...That one sucked, didn't it...
Enough said.
-
Any Chuck Norris fact can be applied to Teddy Roosevelt with the words "an infinite amount of times" added to the end.
That thing's going in 100% original.
-Daniel O'Brien
The 5 Most Badass Presidents of All-Time (Cracked.com) [link]
I'll give you the benefit of the doubt and assume that's a coincidence, but either way, I don't want to step on Cracked's toes, since they've been singing Teddy's praises on the net for much longer than I have.
Hmm... Okay, how about this: Some would call the African elephant the largest land animal in the world, and a beast that could wipe out entire villages when enraged. Teddy Roosevelt called them "Practice."
I like this new one! I'll have to tweak it a bit for length, but it got a chuckle out of me.