Not exactly a gift, but the Hudson's Bay Company original 1670 charter required that it pay 'rent' to the British Monarchy for the right to trap/trade in Rupert's Land, paid whenever the reigning British monarch just happened to be 'round Rupert's Land way to collect - two elk heads and two beaver pelts. The 'rent' was only collected 3 times: 1939, 1959 and 1970, but live animals were substituted in 1970 (which the Queen donated to the Winnipeg Zoo).

I think it was elsewhere on this sub that I first read about Thomas Jefferson's quest to give the French a moose.

It was a scientific mission, because there was a theory going around Europe at the time that all the New World's fauna were small and degenerate, so the lands weren't suitable for breeding big, strong, leader-type men either, which was a problem for someone like Jefferson trying to put forward a general impression of being sensible, vital, economically sound and basically the way of the future.

So when Count Georges-Louis Leclerc Buffon wrote his Histoire Naturelle and said that America was a miasmal place where creatures became enfeebled by the cold swamps, Jefferson knew he had to prove that the native animals were not tiny or weak.

He sent Buffon a panther skin, then mastodon bones. Didn't work.

It had to be a moose, and he had to deliver it personally.

Quoting now from that U Chicago excerpt of Lee Alan Dugatkin's Mr. Jefferson and the Giant Moose:

Both while he was being chased from Monticello by the British in the early 1780s, and then later while he was in France drumming up support and money for the revolutionary cause in the mid-to-late 1780s, Jefferson spent an inordinate amount of time imploring his friends to send him a stuffed, very large moose. In the midst of correspondences with James Monroe, George Washington, John Adams, and Benjamin Franklin over urgent matters of state, Jefferson found the time to repeatedly write his colleagues—particularly those who liked to hunt—all but begging them to send him a moose that he could use to counter Buffon’s ideas on degeneracy. Consider the following letter to former Revolutionary War general and ex-governor of New Hampshire, John Sullivan:

The readiness with which you undertook to endeavor to get for me the skin, the skeleton and the horns of the moose … emboldens me to renew my application to you for those objects, which would be an acquisition here, more precious than you can imagine. Could I chuse the manner of preparing them, it should be to leave the hoof on, to leave the bones of the legs and of the thighs if possible in the skin, and to leave also the bones of the head in the skin with horns on, so that by sewing up the neck and belly of the skin, we should have the true form and size of the animal. However, I know they are too rare to be obtained so perfect; therefore I will pray you send me the skin, skeleton and horns just as you can get them, but most especially those of the moose. Address them to me, to the care of the American Consul of the port in France to which they come.

The hunt for this moose, and the attempt to get it shipped to Jefferson, and then Buffon in Paris, is the stuff of movies. The plotline involved teams of twenty men hauling a giant dead moose through miles of snow and frozen forests, a carcass falling apart in transit, antlers that didn’t quite belong to the body of the moose but could be “fixed on at pleasure,” crates lost in transit, irresponsible shippers, and a despondent Jefferson thinking all hope of receiving this critical piece of evidence was lost. Eventually, though, the seven-foot-tall stuffed moose made it to Jefferson, and then to Buffon.

It only kinda sorta worked. Buffon was suitably impressed, but his degeneracy theory still floated around for a few decades after Buffon and Jefferson were both dead.

I can only imagine what it must have been like for Buffon at the moment he was presented with this much-shipped, slightly modular, cloven-hoofed thing. Did Jefferson try to play it cool? Like, "Oh, yeah, just had the guys send me one of the littler ones, you like it?"

Or was he more "Behold! The MOOSE!"

"Qu'est ce que c'est?"

"THE MOOSE, mon ami. Comme une deer, only bigger!"

In the 1960s, British Prime Minister Harold Wilson visited President Lyndon Johnson at the White House.

As the prime minister was leaving, he gave Johnson a beautiful Burberry coat.

"President Johnson opened the box and put the coat on, and the sleeves came about halfway on his arms," said Ambassador Lloyd Hand, who was the U.S. chief of protocol at the time. "He said 'Lloyd, see if you can catch the prime minister and tell him this is the wrong size.' "

source: http://www.npr.org/2011/05/25/136651998/the-art-and-artlessness-of-the-presidential-gift

While not strictly a history textbook, Danubia is quite a delight to read, although best considered a personal travel memoir of the author Simon Winder.

As the author ambles along the Danube river, he remarks on several rather astonishing gifts through out history, which includes a crocodile in Brno -- gifted from the Turks to Emperor Matthias in the 1600s -- and a bag of Turkish noses from the Uskoks to Emperor Charles V in 1532.

But wait, this isn't the only somewhat unusual gift that Charles V obtains from his many many subjects. Among those more cherished is perhaps the Maltese falcon, a sign of feudal loyalty from the Knights of St. John nèe Hospitallers for the grant of Malta to be their operating base.

In 1763, during the siege of Fort Pitt, it is speculated that the Natives were given blankets infected with smallpox.

Not sure if this counts as a gift but Lincoln was offered elephants by the King of Siam and given a sword and some elephant tusks. Lincoln's letter to His Majesty is masterfully diplomatic about refusing the elephantine offer.

Admittedly, I'm not sure how you'd ship elephants to the US in that time...or have them procreate...or really anything as to the possible merits of elephants in America.