This may seem like an easy question to answer - of course it exists. I can see it and feel it so it must be real. But how do you know that you are not just a brain in a jar, dreaming about the world around you? How do you know that your consciousness isn't just an alien computer simulation being fed fake input about your environment?
These two scenarios are pretty unlikely, since for you to be a literal brain in a jar (or maybe just lying in hospital in a coma) there has to be other people who put you in a jar. It is more probable that you are real as long as there are more people out of jars than in them. For the computer simulation, some civilisation would have had to build and program the computer, and it is unlikely that there are more simulations than people.
However, I think these scenarios can become much more likely if you simply reduce the time the created consciousness exists.
There is no way to tell if the universe has existed for 13.7 billion years or if it was created an hour ago, in the state it was in an hour ago, and with fake memories of the universe prior to its creation in your head. The same is true for the universe being created a minute ago. Or ten seconds ago. Or a single second. The less time that you have existed, the less you experience of your environment and the less memories you access.
In a second you are a aware of very little of the world around you, and so hardly any information is needed to give you the illusion of a continual existence. What about such a small amount of time as a hundredth of a second? In that small instant your senses are practically receiving a static input; your eyes see a still image like a single frame in a movie, your skin feels constant pressure from any objects it is in contact with. Barely any information is needed to represent your sensory input and the memories you access in that short space of time.
If such a small span is enough to give you the illusion of a longer existence, then I think it is logical to conclude that it is more likely than not that this is in fact what we are actually experiencing. Wouldn't it be more probable that the information needed to simulate a few milliseconds of consciousness would occur naturally in some chaotic system than an entire universe being created capable of supporting complex life? The paradox is that if this universe actually existed in the form that it appears to, such instantaneous consciousnesses would be constantly popping up in the tumultuous hearts of stars or the boiling chemical soups of alien seas.
So that's why I think the universe doesn't exist, and as a consequence none of you exist. Of course, from your perspective, I don't exist and you are the single consciousness. I'll read through this post and fix any grammatical or logical errors tomorrow, it's 1 in the morning and I'm pretty tired. This is basically the sort of thing that literally keeps me up at night.
TL;DR How do you know that your mind wasn't created a fraction of a second ago, and will cease to exist soon after?
Anyway, do I exist?