Naming the world's largest organism is complicated by what you define as an "organism". Certain invertebrates, such as coral and fungus, aggregate into collectives called superorganisms. Hence, the world's largest superorganism, the Great Barrier Reef, sometimes gets the title.
There is also a fungus called Armillaria ostoyae, or the honey mushroom, that forms fungal mats as long as 8.9 km² (2200 square acres or 3.5 square miles), although there is disagreement as to whether the entire mat qualifies as a single organism. The largest example of this fungus can be found in a national forest in Oregon. It is estimated to be 2,400 years old.
If you ask which obviously individual largest organism receives the title, it falls to the Giant Sequoia called General Sherman, which can be found in California. This is generally accepted at the world's largest organism. It has a volume of 1,489 cubic meters, or 52,583 cubic feet. Despite this, pages referencing the Oregonian honey mushroom rank highest for Google searches for "largest organism".
Mushrooms and trees aside, the world's largest animal ever known to have lived is the blue whale, whose heart is the size of a Volkswagen bug. These underwater giants weigh up to 180 tons and can be as long as 30 meters (100 feet). At birth, they are the size of a fully grown hippopotamus. Because so many blue whales were killed in the early 20th century in the Antarctic by sailors unversed in scientific measurement techniques, we are not entirely certain how large blue whales can really grow, but they are definitely massive.
The blue whale is only rivaled in size by Argentinosaurus, the largest and heaviest land animal to ever have lived, with a height of more than 40 meters (125 feet) and a weight around 90 tons. Argentinosaurus was an herbivorous sauropod, the largest of its kind that we know of. We have only found a few bones - a single vertebrate was 1.3 m (4.3 ft) long. While not the world's largest organism in general, its appearance in the flesh could probably impress humans even more than any of the aforementioned organisms. Argentinosaurus lived 112.2 to 93.5 million years ago.
Repeatedly, throughout the summer of 1997, an incredibly loud, low-frequency sound was detected by the Equatorial Pacific Ocean autonomous hydrophone array. The sound is very perplexing because it has the acoustic signature of an immense organism with a gas-filled sac. This has since become known as the Bloop. It has been hypothesized this is a new, larger form of whale, but it would need to surface every once in a while, which would make it visible occasionally. So the Bloop remains a mystery, but it could end up being the call of the world's true largest organism, or at least its largest animal.
One day we may discover an even larger organism. Life on other planets or genetically engineered life probably has a size limit substantially larger than anything we have ever seen or imagined.