What is the largest Living thing?

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It’s a mushroom.

It’s not even a particular rare one. You've probably got the honey fungus in your garden, growing on a dear tree-stump.

It covers 890 hectares and is between 2000 and 8000 years old. Most of it is underground form of a massive mat of tentacle-like roots. These spread along tree roots, killing trees and peeping up through the soil occasionally as innocent-looking clumps of honey mushrooms.

The Giant honey fungus of Oregon was initially thought to grow in separate cluster throughout the forest, but researchers have now confirmed it is the world’s single biggest organism, connected under the soil.

Reference:
Lloyd, J., & Mitchinson, J. (2006). The book of general ignorance. New York: Harmony Books.