Fully Homomorphic Encryption [1] (FHE) would be one of these services. You could let someone else on the internet do computations for you, without him being able to see what the input/output of these computations are. The idea behind FHE is that the whole computation process will be fully encrypted, and therefore anonymous, while containing usable personal data. From the server side you can apply algorithms like:
// With FHE the server side cannot see the outcome of the if/else statement.
if (encData.authenticate('John Doe')) {
encData.result = addEncryptedResult(doStuffWith(encData.input), encData.pubKey);
} else {
encData.result = addEncryptedResult("Authentication failed.", encData.pubKey);
}
sendBack(encData);
For example, you can use FHE on your income. Send the encrypted data to someone else (could be anyone). They compute the tax reduction computations for you, without seeing who you are, what you earn, and what your tax returns are. Because every computation this person performs is fully encrypted and release the results openly. You download the encrypted results and, because you have the private key, find your tax returns. (I can probably come up with a better example if I think a bit longer)
The only downside of FHE is that it's way too slow to be practical. When I studied the Brakerski-Gentry-Vaikuntanathan scheme in 2011 I remembered that it took hours to compute 3 + 4. Speed has increased a lot in FHE, but it's not at a practical level yet.
[1] - https://en.wikipedia.org/wiki/Homomorphic_encryption