For the largest number, that's a tricky one. There is no limit to the amount of numbers one could come up with, as you can always add 1 digit anywhere inside of a number and it becomes bigger. This is where infinity comes in.
Infinity is a concept, meaning it isn't a real number, however we do use it in the math world a lot. infinity is the thought that there is no end, for example pi is an infinite number (meaning there is an endless amount of digits.) if you hit 9 on a keyboard and held it forever, you still wouldn't reach the biggest number in existence, because you can always add another 9.
For the largest thing, That moreso depends on what requirements something has to meet for it to be the 'largest' in your eyes. A blue whale is the largest animal, Annapurna is the largest mountain, and so on, but I think the largest thing in the world is the world itself, because the world is a thing. Neither of these questions really have a true "answer," but I hope I could explain the concept of infinity well.