Wondrous mathematics: Large numbers, very large numbers and very very large numbers

This talk is for all who enjoyed the game "who can name the larger number?" as a kid. This talk takes you on a tour of the wondrous world of mind-boggingly large numbers. In case you are new to the business of extremely large but still finitely large numbers, be prepared to be in thorough awe at hyper operators and Graham's number, a number so large not even the number of its digits fits into our universe. In case you've been a longtime follower of Graham's number, be prepared to be amazed by numbers which render Graham's number tiny and insignificant in comparison. Some of the numbers we present go beyond the boundaries of computation. Some even go beyond the boundaries of logic, while still staying clear of paradoxes, and some require stronger and stronger philosophical commitments. We will also present reasons why mathematicians are interested in very large numbers. There will also be a companion talk on infinitely large numbers. This talk is not a prerequisite for the other, and vice versa. Over the course of the first three days of congress, we also run a large number contest. We invite you to participate in this contest. The award ceremony for this contest is part of this session. Details will be announced later.