Background
Recently, I have been writing up some notes on derived functors and I came across the Eilenberg-Watts theorems [1], which essentially explain why it is hard to find derived functors besides Ext and Tor. Before asking my question, allow me to briefly state these theorems.
Let $R$ and $S$ be rings and $M$ and $S-R$ bimodule. A basic property of the functor $M\otimes_R-:R-\mathrm{Mod}\to S-\mathrm{Mod}$ is that it is an additive covariant right-exact functor. In fact, it also commutes with direct sums. Curiously these properties are enough to characterize it: any covariant additive $T:R-\mathrm{Mod}\to S-\mathrm{Mod}$ that is right-exact and commutes with direct sums is in fact naturally equivalent to some $M\otimes_R -$ for some $S-R$ bimodule $M$. This is the statement of the Eilenberg-Watts theorem for tensor functors.
For completeness, I should state the other version for Hom. If $T:R-\mathrm{Mod}\to S-\mathrm{Mod}$ is an additive left-exact contravariant functor which converts direct sums into direct products (i.e. $T(\oplus M_i) \cong \prod T(M_i)$) then there is an $R-S$ bimodule $M$ such that $T$ is naturally equivalent to $\mathrm{Hom}_{R}(-,M)$.
Finally, if $T:R-\mathrm{Mod}\to \mathbb{Z}-\mathrm{Mod}$ is left-exact covariant that commutes with inverse limits then there is a left $R$-module $M$ such that $T$ is naturally equivalent to $\mathrm{Hom}_{R}(M,-)$.
The Prompting for the Question
From the above, any functor satisfying the hypotheses of these Eilenberg-Watts type theorems are going to be naturally equivalent to either a tensor or a Hom, and thus its derived functors will just be Tor or Ext respectively. For instance, if $G$ is a group then $G-\mathrm{Mod}\to \mathbb{Z}-\mathrm{Mod}$ given by $A\mapsto A_G$, where $A_G$ is the quotient of $A$ by the submodule generated by $ga – a$ for all $g\in G$ and $a\in A$ is just the usual coinvariant functor, whose left derived functors are the homology groups $H^i(G,A)$. By Eilenberg-Watts, $-_G$ must be equivalent to some tensor functor, and in fact it is easy to prove that $A\mapsto A_G$ is naturally equivalent to $A\mapsto \mathbb{Z}\otimes_{\mathbb{Z}G}A$
Incidentally, the proof given by C.E. Watts is explicit enough so that the above natural equivalence is apparent.
The Question
Notice that in each of these the hypothesis of playing nice with limits is required. I am actually interested in functors which do not play nicely with limits. For instance,
What are some examples of covariant right-exact functors $T:R-\mathrm{Mod}\to S-\mathrm{Mod}$ that do not commute with all direct sums? [Edit: $T$ also should not be left exact in this case.]
Such a $T$ of course cannot be a left-adjoint for otherwise it would commute with direct sums. Such a $T$ could be interesting because its left-derived functors may not be "like" the Tor functor. The question also goes for dropping the playing-nice-with-limit hypotheses in the other forms of the theorem. I tried a Google search but could not seem to find anything relevant.
Since I am asking for a list of examples, I have made this a community wiki. Thanks!
[1] Watts, "Intrinsic Characterizations of Some Additive Functors". Proceedings of the American Mathematical Society, Vol. 11, No. 1 (Feb., 1960), pp. 5-8
Addendum (edit)
Thanks everyone for their answers; I think I should have been more precise and asked a question more along the lines of:
What are some derived functors that are not Ext or Tor?
Which I believe some of the existing answers are. In essence I wanted examples that were neither tensors nor Homs in disguise…
Best Answer
Here is a specific example, though it admits obvious generalizations. Let $R=S=\mathbb{Z}$, and consider the functor from abelian groups to abelian groups defined by $$T(X) = \mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Q}_p/\mathbb{Z}_p,X),$$ where $p$ is a prime. This is right-exact since $\mathrm{Ext}^2_{\mathbb{Z}}\equiv0$. It does not commute with direct sums, a fact which is clear when you observe that $T$ is also the $0$th left derived functor of $p$-completion, and $p$-completion does not preserve infinite sums, even when applied to free abelian groups.