Is there a way to keep track of an infinite list of numbers without taking up too much "room"?

Question

This started out as me pondering what it would be like to live forever. I realized if one was to live forever that you would still want to keep track of time and what the date was. Eventually though, if you kept track of the year you would eventually get to a year that would be too long to write down efficiently, for example living long enough that in size 12 font that the year would take enough pages to write down that setting them side by side would be over a mile long. Obviously, with enough time having passed this would happen eventually. So I am curious, is there a way to keep track of the time without running into a problem such as this, or without a possible loss of information (for example resetting the year after every 2000 years back to 0). I thought of using continuous objects and if one were perfect it seems they could write down the year as a sin curve and use slight variations in the curve to represent different large numbers. I'm aware getting used to looking at these would be difficult, I am not looking for something that is comfortable just something that works. I'm guessing this is some kind of information problem in which I'm asking if it is possible to keep an infinite amount of information such as a date in a convenient form to interpret without taking up infinite space trying to do so. The way you interpret the new "number" casually does not have to be nice, it would be enough just to be able to look at 2 objects and see that one is bigger than the other and that there is a process to convert them backwards to a natural number. I like the idea of converting them to graphs of continuous functions because they are easy to change minutely in a small space without changing the graph too much but I'm open to other possibilities if there are. The sin function was purely personal preference.