Here are four claims:
(1) Any outcome of any action is either more valuable than, less valuable than, or equally as valuable as any other outcome of any other action.
(2) When choosing among possible actions, one ought to do that action whose outcome has the greatest value.
(3) When choosing among possible actions, it is one’s duty to do that action whose outcome has the greatest value.
(4) When choosing among possible actions, one ought most to do that action whose outcome has the greatest value; one ought least to do that action whose outcome has the lowest value; one ought more to do an action whose outcome has higher value than one with lower value; etc.
I think that a person who believed none of these claims could not properly be called a consequentialist. Many consequentialists, I think, believe all four of these claims, but I take claims (1) and (4) to be the most "central" to consequentialism. (2) may follow from (4). (3), I think, does not follow from the others. In the course of this post I will try to give reasons to think that consequentialists should to reject (3).
Actions are "supererogatory" when they "go beyond the call of duty." If moral reality were a high school class, then supererogatory actions would be done for extra-credit points. You wouldn’t lose points if you failed to perform these actions; you could pass the class without ever once performing one. But you’d certainly be encouraged to perform them, and all the best students would perform them.
Sometimes it is said that consequentialists are unable to classify any actions at all as supererogatory. This would be bad for consequentialism, since supererogation seems to occupy a central place in everyday moral reasoning. I think (3) is likely to commit consequentialists to this bad result, but I think that if consequentialists reject (3), then they can avoid this bad result.
Consider the following example. Suppose that, after Monica has budgeted for groceries and other necessities, 30% of her annual income is unaccounted for. Monica is trying to decide how to spend that 30%. She has four choices:
(a) Give the money to a charity.
(b) Spend the money on an expensive car.
(c) Throw the money into a river.
(d) Give the money to a fascist dictator.
Let’s assume that these choices are presented in descending order of value, so that if Monica wants to maximize the amount of value resulting from her action, she will choose to do (a). In that case, if (3) is true, then doing (a) would be Monica’s duty. And in that case, there does not seem to be any room for supererogation. For the only possible supererogatory actions left are (b), (c) and (d). But these actions all result in less value than would the action we have identified as Monica’s duty. It seems highly unlikely that one could go beyond one’s duty by doing something with a outcome than one’s duty itself.
Suppose, then, we reject (3). Then Monica’s duty is not necessarily to do (a). Perhaps she has no duty. I think this would be consistent with (1), (2) and (4); after all, those claims do not mention duty. Or perhaps Monica does have a duty — perhaps her duty is to do (b). This is also compatible with (1), (2) and (4). In that case Monica would be going beyond her duty to do (a), since doing (a) would result in greater value than doing her duty itself, i.e. (b). So, by rejecting (3), it appears that room is made for supererogation, at least in the example just discussed.
What would a consequentialism without (3) look like? Obviously, it would be a consequentialism according to which one’s duty is sometimes to do something other than maximize value. This might sound like a non-consequentialist theory. But I would still want to classify it as a consequentialist theory if it held on to (1), (2) and (4). For if it held on to those claims, then it would say that one could always morally do better by maximizing value, and it would say that one always ought most to do whatever must be done in order to maximize value.
There appears to be an inconsistency between (2) and the denial of (3). For if one always ought to maximize value, as (2) says, then how can it be that one’s duty is not always to maximize value, as not-(3) says? I think it is true that one ought always to do one’s duty. But I think it is also true that one ought always go beyond one’s duty. "Duty" seems to be a kind of moral "zero-marker." Any action "below" one’s duty is "negative," and ought not be performed; any action "above," or "at," one’s duty is "positive," and ought to be performed. This means that if one always ought to maximize value, then maximizing value is either one’s duty, or is beyond one’s duty. So, if one holds (1), then one can consistently deny (3), as long as one continues to classify actions which maximize value as beyond duty.
E.G. 
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