Dear Tim: I am not sure I understand your question. I think any reasonably technical field has a few examples. I heard algebraic geometry is such a field, so I looked up a couple of sources and found these two right away:

1) Let $f: R\to S$ be a local homomorphism of local rings. Suppose $R$ is regular, $S$ is Cohen-Macaulay, $f$ is finite and $\text{dim} R = \text{dim} S$. Then $f$ is flat. 

This is a basic result that is used quite frequently. I don't think you can drop any of the hypotheses, they are all basic definitions and independent of each other. 

Or how about:

2) A morphism of schemes $f: X \to S$ is quasi-finite if it is locally of finite type, quasi compact and has finite fibres. 

Again, I don't think you can drop any of the hypotheses. I am reasonably sure EGA has a few more results like this (-: