Non-Universal Delta Functors

Recall a delta functor on an abelian category is a collection of functors $H^n$, $n \ge 0$ and connecting homomorphisms associated to each short exact sequence so that we get a corresponding long exact sequence in cohomology (http://en.wikipedia.org/wiki/Delta-functor). If we have enough injectives, then the derived functors of a functor $F$ create a universal Delta-functor, where maps to other delta functors are uniquely determined by the maps on $H^0$.

My question is what are some interesting examples of non-universal delta functors for a fixed $H^0$? Clearly we can for example take shifts of the universal one and direct sum them to the original one (after applying some exact functors perhaps). Phrased another way, are there any genuinely different Cohomology theories, or can we somehow "create" all others just using the universal one? If this is too general, can we say anything in cases pertaining to Algebraic geometry (e.g. $O_X$-modules on a variety $X$ say).

Motivation: The only time I know cohomology theories to produce long exact sequences are when we can prove they are, in certain cases, the same as is calculated with a derived functor.

There are genuinely different cohomology theories. Here is an example. We can show that there is a (nonnoetherian) ring $A$ with a multiplicatively closed subset $S$ and an injective $A$-module $I$ such that $S^{-1}I$ is not injective over $S^{-1}A$.(See:E.C.Dade, Localization of injective modules, J. Algebra 69 (1981), 416-425; F. )

Now that $S^{-1}I$ isn't injective, there is a $S^{-1}A$-module $M$ such that $Ext^1_{S^{-1}A}(M,S^{-1}I) \neq 0$. We may consider the $\delta$-functor $Ext_{S^{-1}A}(M,S^{-1}\rule{0.5cm}{0.15mm})$. This clearly isn't universal and it isn't of the form you described, as it agrees with the universal $\delta$-functor for $S^{-1}A$-modules when considered as $A$-modules, but not in general.