The problem with that is on the other practical end: if you massively increase the resources needed will also increase the server side resources; it won't be as bad as it will be on the cracking end, but server resources are expensive.
It won't be as bad as on the cracking end, that's the whole point. The reason for doing password hashing is to exploit the asymmetric level of effort between hashing and brute force search. To make that work, though, you do need to invest as much as you can afford in the server, to move the bar for the attacker as high as possible -- hopefully out of reach of all but the most serious. If you can't afford very much, fine, but realize that you're also not setting the bar very high.
But this is exactly why good password hashing algorithms are moving to RAM consumption as the primary barrier. It's pretty trivial for a server with many GiB of RAM to allocate 256 MiB to hashing a password, for a few milliseconds, but it gets very costly, very fast, for the attacker. And if you can't afford 256 MiB, how about 64?
What you definitely do not want is a solution that takes microseconds and uses a few dozen bytes. That creates a trivial situation for the attacker given modern hardware, and your server absolutely can afford more than that.
This is similar to why we don't use much longer keys for public key encryption and use really large primes for DH key exchange.
Nope. The leverage factor in the password hashing case is linear, since the entropy of passwords is constant (on average). The leverage factor for cryptographic keys is exponential. The reason we don't use much longer keys for public key encryption, etc., is because there's no point in doing so, not because we can't afford it. The key sizes we use are already invulnerable to any practical attack in the near future. For data that must be secret for a long time, we do use larger key sizes, as a hedge against the unknown.