Is there possibly a largest prime number? Prime numbers are numbers with no factors other than one and itself.

Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of "possible factors" that number might have.

So the larger the number, it seems like the less likely the number is to be a prime.

Surely there must be a number where, simply, every number above it has some other factors.  A "critical point" where every number larger than it simply will always have some factors other than one and itself.

Has there been any research as to finding this critical point, or has it been proven not to exist?  That for any $n$ there is always guaranteed to be a number higher than $n$ that has no factors other than one and itself?
Euclid's famous proof is as follows: Suppose there is a finite number of primes.  Let $x$ be the product of all of these primes.  Then look at $x+1$.  It is clear that $x$ is coprime to $x+1$.  Therefore, no nontrivial factor of $x$ is a factor of $x+1$, but every prime is a factor of $x$.  By the fundamental theorem of arithmetic, $x+1$ admits a prime factorization, and by the above remark, none of these prime factors can be a factor of $x$, but $x$ is the product of all primes.  This is a contradiction.
Proof that there is always a higher prime number and therfore an infinite amount of prime numbers:

Mathematicians often hide behind an array of jargonish symbology. We need more proofs like the one I invent here (took me 90 minutes after watching “An infinite Mind” thanks Patel for channeling Ramanujan):

Easiest layman’s proof that there is always a higher prime number than any prime you are given:

Let Y be a prime number of any positive value.

Y!-1 cannot be divided by Y or any other integer lower than Y because those numbers are already factors of Y! and therefore cannot be factors of Y!-1; Therefore any factor of Y!-1 must have a value higher than Y, and yet not be divisible by any other factor lower than Y, which means it must be a prime number higher than Y. So in any case, either Y!-1 is a prime number OR Y!-1 has at least one factor which is prime numbers higher than Y: in either case, there is a higher prime number than Y.  

[Y! means 1x2x3x4x5x … all the way up to … Y]