How many kinds of infinity are there?

Types of infinite numbers and some things they apply to:

Cardinals (set theory, applies to sizes of ordinals, sizes of Hilbert Spaces)
Ordinals (set theory, used to create ordinal spaces, and in ordinal analysis. Noncommutative.)
Beth Numbers (like Cardinals, or not, depending on continuum hypothesis stuff)
Hyperreals (includes infinitesimals, good for analysis, computational geometry)
Superreals (maximal hyperreals, similar to surreals)
Supernaturals (prime factorization matters, used in field theory)
Surreals (Best and most beautiful thing ever, maximal number system, combinatorial game theory)
Surcomplex (surreal version of complex numbers)
Infinity of Calculus (takes things to limits)
Infinity of Projective Geometry (1/0=infinity, positive infinity equals negative infinity)
Infinite Hilbert Space (can be any Cardinal number of dimensions)
Real Line (an infinite line made up of all real numbers)
Long Line (longer than the real line, in topology)
Absolute infinity (self-contradictory, not really a thing)

Non-infinite kinds of numbers:

P-adic (alternative to real numbers)

Natural numbers (1, 2, 3…)
Integers (…-3, -2, -1, 0, 1, 2…)
Rationals (1, 1/2, 2/1, 2/3, 3/2, 3/4, 4/3…)
Algebraic (sqrt 2, golden ratio, anything you can get with algebra)
Transcendental (real numbers you can’t get using any finite amount of algebra, like pi and e)
Reals (all possible infinite sequences of digits 0.123456789101112131415…, includes all of the above)
Imaginary (reals times i, where i^2=-1)
Complex (one part real, one part “imaginary,” a consistent, commutative, associative, 2-dimensional number system)
Dual numbers (instead of imagining a number where i^2=-1, make up a number where ε^2=0 and use that)
Quaternions (make up numbers that square to -1, but are different from each other. i^2=j^2=k^2=ijk=-1. 4d, noncommutative.)
Octonions (make up even more numbers, 8d, noncommutative and nonassociative.)
Split-complex (imagine if i^2=+1, but i isn’t 1)
Bicomplex number, or tessarine
Hypercomplex (category that describes/includes all complexy number systems that extend the reals)

Also see combinatorial game theory, which extends the surreal numbers to get numberlike but not-quite-number values such as “star.” Star gets confused with zero, in a mathematical definition of confusion, but it is not actually zero.

You can also write real numbers in other bases, including negative bases, irrational bases, and even complex bases.