The security of quadruple or even more encryption isn't much more than that of double encryption. But each round also counts as one encryption. So why isn't triple DES only 96-bits secure?
A meet-in-the-middle attack basically halves the number of bits of the total key size, rounding up. So 3 and 4 both have 2x security and 5 and 6 block cipher encrypts provide 3x security of the original cipher. I think that this is the main misunderstanding of meet-in-the-middle attacks.
The DES cipher itself is protected against meet-in-the-middle attacks, mainly due to how the key is expanded. Note that the key material is related and that there are many rounds within a single DES block encrypt, which makes it different from how block ciphers can be stacked on top of each other. You can see a question about meet in the middle attacks within block ciphers here (note who is asking the question).
So it isn't that this isn't an active field of crypt-analysis. Actually, many amateur block ciphers will probably fail against the meet-in-the-middle attacks and are thus rendered insecure (assuming that they specify e.g. 128 bit security for a 128 bit key).