Graham's number is big. So big that it's hard to even grasp how big it actually is. So maybe it's better to start with some "smaller" numbers first to truly realize the magnitude of such numbers.
As a kid thousand things was already a lot and million was the biggest thing ever. Coming up with even bigger numbers always involved just adding more zeroes to the end of the numbers. Million millions and so forth.
Adding zeroes still gets you only so far. The culmination of that method is probably the googolplex. Googol is a number we get when we have one followed by a hundred zeroes or 10^100. Googolplex in turn has a googol zeroes. Pushing a bit further before leaving any hypothetical physical boundaries behind we could come up with a number 10^10^185. That is the biggest number we could write out. That is if we could write each zero within a Planck's volume, the smallest physically space possible. That's all that could fit in this universe.
These numbers are already beyond the cosmological proportions, but still they are just a speck of dust within the galaxy compared to the number's such as the Graham's number.