What is the cost of encrypting of long message with public-key cryptography?
The real problem is not the encryption or even the decryption of an encryption scheme with huge values, it is the generation of the key:

For RSA, ElGamal, Paillier, etc. you need one or more large primes, which make up a modulus for your computations. In practice, you can find large primes by using the Miller-Rabin primality test. It has a run time of $O(t \log^3 n)$, and for large numbers it can be improved to $O(t \log^2 n)$ if using FFT-based multiplication. $t$ notes the number of test runs and you have an error rate of $1/4^k$, where the test says prime for a coposite number. While the runtime with multiplication optimazation has the smaller exponent, the Big-O notation does hide constant factors, and I don't know where the break-even is, but let's just assume the current crypto libraries already use the optimzation.

As a reference in this SO answer generating a $2048$ bit RSA key (which is generating two $1024$ bit primes and a tiny amount of its computation time to find $e$ coprime to $\phi(n)$ and calculating $d$) takes around $1$ sec. So $\approx 0.5s$ for a $1024$ bit prime.

If we want to increase the size by a factor $a$, then the Miller-Rabin test takes $a^2$ times longer. However, we also need to test $a$ times as many numbers on average to find an actual prime number. Example: A random $1000$ bit number is prime roughly with probability $1/ln(2^{1000}) = (\log 2)/ 1000$ and we need on average $1000/(\ln 2)$ tests to find a prime, a random $2000$ bit number is prime with probability $(\log 2)/2000$, and on average we need to test $2000/(\ln 2)$ numbers to find a prime. This means, we expect to run the algorithm on $a$ times as many numbers to find a prime.

For example, for a prime with $1 M$ bits for example and an estimate of $0.5$ sec for a $1024$ bit prime, we have a factor of $~976.5$ in size, so the runtime for Miller-Rabin itself is $0.5 \cdot 976.5^2$, and we need to test $976.5$ as many values, resulting in an overall runtime of $\approx 465 M$ seconds, which is around $14.76$ years.

In case you want to use RSA you need two of these primes, so after 30 years you have a RSA modulus with $2,000,000$ bit, which allows you to encrypt $0.25$ MByte of data in one piece. Of course using a large computing center will speed this up, but the real problem is the size, since I guess you didn't have $0.25$ MB of data in mind when you wrote "potentially very large".

Edit:
The encryption and decryption scale much less from the size increase. For RSA, you can still use a small exponent in the encryption, this doesn't have to be much larger than current ones. That means the number of steps in square-and-multiply are roughly the same, just each multiplication and modulo operation takes more time. Decryption then involves a number of length of the modulus, but overall it is still quadratic at worst, and much less than prime generation.