I am always interested in the 'what if' of history, and this topic is an excellent example. Jefferson's Louisana Purchase more than doubled the size of the United States and abandoned (at least for that moment ) his philosophy of the strict interpretation of the U.S. Constitution. All one has to do is look at the names of the cities of California to realize how different the U.S. would be. San Francisco, Los Angelos, San Diego, were the names of the Spanish Missions founded during the Age of Exploration and engulfed the west coast of America. If it were up to the Spanish they too would own part of what we now call the United States of America. The Louisana Purchase provided the way for American westward expansion, however it was the Monroe Doctrine that provided the means for American westward expansion. The Louisana Purchase was the greatest land deal ever made, but it would take a U.S. foreign policy to enforce not only westward expansion, but what I think of as American Expansion. On the world stage, The Monroe Doctrine was probably the first document since The Declaration of Independence that defined, or for that matter, redefined the who, what and where The United States was. If it were not for the Louisana Purchase and the later foreign policy of the Monroe Doctrine the United States would have probably found itself forever tied to the quagmires of the Spanish and French. With all due respect, Lord knows how difficult that would end up being...
If the Lousiana Purchase had not happened, the obvious effect would have been a slower westward expansion by the US. Given that the area would not have been US territory, the historical narrative would have probably developed something along the lines of the settlement and incorporation of Texas, which had been Mexican territory. The inevitable European crush of immigration in the 1800's pushing westward would have populated the area with settlers seeking to ally themselves with the US -- only because by the early 1800's both Spain and France had declining political power in North America -- and Britain had lost her colonies. If the Lousiana Territory had remained French or Spanish, as it had been a few years before France acquired it, there may have been a few border skirmishes, as there had been between France and England during early colonial times, but little else would have changed. It certainly would not have remained French; as #3 asserts, Napoleon sold it to raise money to continue his conquest of Europe.
If the Louisiana Purchase had not taken place, the United States would not be one country from coast to coast. We would have a territory of France in the middle southern section of the United States. That territory would have a separate government, with its own laws, military, and law enforcement. A French territory would not be subject to the United States government at all. Trading between the French territory could only happen through treaties between the United States and France. To travel through the territory to go to a part of the United States on the other side, one would need a passport and a visa. Since the United States and the territory would share the same Gulf Coast, many issues could arise, for example, the issue of territory for fishing, the issue of border protection, and the issue shipping. Presumptively, the people in a French territory would speak French, thus requiring people along the border to learn French, too, or requiring the need of translators for many situations. There are, I am sure, other implications, but this should be enough to get you started.
We can only speculate on how the United States might be different, of course, but there is no denying the fact that the acquisition of the Louisiana Purchase brought a sizeable chunk of real estate laden with French influence into the nation. Had the French maintained control of this area, it seems likely that Britain might have been even more interested in re-establishing the North American control that they were after eleven years later, in 1812, when the so named War of 1812 resulted in the burning of Washington, D.C., among other things. How much France might have been able to do to defend the Louisiana territory at that point is questionable; Napoleon actually sold Louisiana to raise cash at the turn of the century.
The United States would be very, very different from what it is today if the Louisiana Purchase had not happened. The Louisiana Purchase not only doubled the size of the United State's territory, but also contributed to the mentality of constant expansion in the United States throughout its history.
If the Louisiana Purchase had not happened, the territory of the United States may still be limited to east of the Mississippi today. The Purchase not only added a huge expanse of French territory to the U.S., but also allowed for U.S. expansion into land all the way to the Pacific Ocean. The land of the Louisiana Purchase lay between the original colonies and the Pacific Ocean. Now that the land in between was in their hands, American could access the far West of present day U.S. territory.
Also, the Louisiana Purchase facillitated Westward expansion, which created the mentality of constant expansion in American society. The frontier's contribution to the United States was not only economic. The frontier created an atmosphere of pioneering energy throughout the United States that surely contributed to its development, as well as to the nationalism of the American people.
The Louisiana Purchase threw the doors to the land west of the Mississippi wide open for the Americans.
If the Purchase had not gone through, with France in the middle would have led to a faster rise of militarism in boith France and the US and led to many conflicts and wars over the territory. I believe that the US would finally prevail, defeating the French for expansion to the Pacific, but in the end, end up with French and English as the national languages.