Larger than infinity

We are taught that infinity is the largest possible number, but does that mean if something is infinitely large something else cannot be larger?  I have pondered this occasionally for many years since it first occurred to me during grade school.  Perhaps infinity should be allowed to be raised to powers and expressed exponentially.

Take a one dimensional line as an example.  Although it has no width, it contains an infinite number of points.  Conventional thinking would dictate that there is no larger number than the number of possible points on this infinite line.  But what if we compare it to a plane?  Now that we have added a second dimension that also extends infinitely, there is an infinite number of points above and below every point on the original line.  Would it not be accurate to refer to this as infinity squared?  We could then add a third dimension and consider a cube that extends infinitely in all three directions.  Now, for every point on the plane there is an infinite number of points in front and behind it.  Would this not be infinity cubed?  As we continue this thought exercise it becomes much more difficult, but it would seem logical to progress through a higher exponent for each additional dimension, each one being infinitely larger than the dimension before it.  We could consider naming this concept beyondfinity, to contrast it with infinity.

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