Why do we need to apply more force to push a submerged object deeper?

The force required to push an object into water increases as the object submerges, i.e. as the amount of water the object displaces steadily increases. But I think if you do the experiment carefully you will find that, once the object is fully submerged, the force required should be almost constant.

Thereafter, many objects get easier to push down with increasing depth, as the water pressure crushes them and they therefore displace less water. Wetsuits, for example, become greatly less buoyant with depth for this reason, which is why divers usually wear a buoyancy compensator.

At extreme depths, if something is less compressible than water, it will become harder to push down owing to the increasing density of water with depth. Factors such as this are important in the design of deep sea submersibles and bathyscaphes such as Alvin and the Trieste.

Well, firstly, I'm not a physics pro but here is a possible explanation.. When you talk about pushing a bucket into water.. I guess initially YOU were not submerged so there was no buoyant force on your body. However, as you pushed the bucket deeper, you would have stepped into the water so you yourself would experience an upward buoyant force. This would make it difficult for you to push the bucket deeper down. Or it might have something to do with the portion of the bucket submerged.. as the answer above this explains..