Look out now, take care, beware of endless rooms there...stretching on forever at Infintiy Hotel

The Infinity Hotel was built inside the German
mathematician David Hilbert’s brilliant’s mind and
revealed to a wider public at the second
International Congress of Mathematicians (1900) as the
first of 20 mathematical problems .

Hilbert used the analogy of Infinity Hotel as a place
with an infinite number of rooms and an infinite number
of guests in those rooms - the hotel can always
accommodate a countably infinite number of new guests
should they turn up.
Hilbert built his Infinity Hotel, by way of a thought
experiment, to demonstrate simply just how weird and
counterintuitive infinity is.
 Infinity Hotel is where mathematicians, philosophers and
scientists sometimes go to conduct thought experiments

Once you’ve visited Hilbert’s Infinity Hotel it’s one of
those places that will keep you coming back.
But be warned, for this hotel is not so safe and infinitely
 accommodating as it might seem. For it is in infinity.
Aaaaagh!
Don't worry, you can always check out.