In my opinion, the greatest discovery in my lifetime will be the discovery of life on another planet/moon. Although I doubt we'll discover intelligent life in my lifetime, I'd bet good money we discover some type of life, such as simple cell organisms in the next decade or two. We've already found several forms of life living in very difficult conditions, which you can read more about here.
Extreme survival: The toughest life forms on Earth
In addition, we've found extremely strong evidence for large amounts of liquid water on Mars' surface in the past. Although we've known for some time that liquid water existed on Mars sometime in the past, NASA recently announced that enough water to exceed that of the Arctic Ocean on Earth existed at one time.
NASA Research Suggests Mars Once Had More Water than Earth’s Arctic Ocean
Although no past life on Mars has been found, where there's water, there is the possibility of life.
Mars isn't the only place in our Solar System where liquid water exists or existed. The moons Europa, Enceladus, and Ganymede most certainly have liquid water underneath their surfaces today. Many other moons in the Solar System likely have water as well.
Jupiter's Moon Ganymede Has Salty Ocean with More Water Than Earth
On top of that, life may exist in other forms where liquid water doesn't exist. Saturn's large moon, Titan, has liquid methane and ethane seas/lakes. Life of some form could exist there.
Cassini Zooms in on Sunny Hydrocarbon Seas and Lakes on Titan
The opportunities for life elsewhere are limitless! The chances of life on any specific object are very slim, but given the billions and billions and billions of planets and moons in our galaxy (and billions of galaxies) it is all but certain many forms of life exist. We just have to find that life.
As I said above, it will happen. We will discover life elsewhere. I think it will happen in the next two decades, but maybe it will take longer. I only hope it happens in my lifetime. If not my lifetime, then the lifetime of my children and grand children. I want them to experience this just as much as I want to experience it.