What do you think are the strongest objections to Consequentialism? (By 'Consequentialism' I roughly mean the unconstrained pursuit of the good -- which might be agent-relative, but shouldn't build in intrinsic concern for traditional "side constraints" like promises, fairness, etc.)
* Counterexamples: I've previously explained why I'm not impressed by the standard "counterexamples" to consequentialism (transplant, bridge, etc.). In short, they involve situations where the supposedly "consequentialist" act seems morally reckless, and merely stipulating that it "really is" for the best predictably doesn't undo our intuitive aversion to such irresponsible behaviour. I think it's a lot harder than most people realize to come up with a real case where an act both (i) maximizes rationally-expectable value, and yet (ii) seems morally repugnant on reflection. So I wish it weren't so common for people to breezily dismiss Act Consequentialism with a mere hand-wave towards the "familiar counterexamples".
Exception: cases of sadistic mobs, etc., which instead call for axiological refinements (reject hedonism!).
* The Separateness of Persons: This venerable objection is vulnerable to the observation that it erroneously assumes that commensurability entails fungibility. (That is, assuming we understand normative 'separateness' as a matter of being non-fungible, distinct values or desirable ends. Sometimes it is instead used in a contentless way to merely emote disapproval of consequentialist aggregation, or to reassert using new words the trivial observation that consequentialism is incompatible with non-derivative rights.)
* Objections to Aggregation: The critic does better, I think, to just object directly to the idea of aggregating welfare. And I agree there are cases where it seems highly counterintuitive to let many small benefits or harms outweigh one great one. But I think considerations of iteration show that our initial intuitions are simply mistaken here.
* The Demandingness Objection: It's often not clear that this is even an objection, as opposed to just a complaint. (However rough on us it is to have to pony over some hard-earned cash, it would be far rougher for the child with intestinal parasites to have to go on untreated! So what, exactly, are we supposed to be complaining about?)
A more sophisticated variant of this objection claims that utilitarianism commits one to implausible character evaluations (e.g. that we're all moral monsters). But this is not so. Combine with deontic pluralism: Act Utilitarianism is an account of the ought of most reason, not the ought of minimal decency. (This move also allows consequentialists to recapture the category of the supererogatory in a plausible way, for those who care about that. See my 'Satisficing' paper for more detail.)
* The Nearest and Dearest Objection: A related (but importantly distinct) objection holds not that utilitarianism is too demanding, but that it demands the wrong things: we should care more about promoting the welfare of our loved ones in preference to strangers. Maybe! I think one could reasonably go either way on this one. But at most it gives a reason to move to a form of agent-relative consequentialism. (Which I don't think makes a huge difference in practice, so long as you give any non-trivial weighting to the welfare of distant others.) It doesn't do anything to motivate deontological side-constraints.
* The Self-Effacingness Objection: is generally no objection at all, except for my souped-up version which can be answered. (Note that the picture of the rational utilitarian agent which emerges from that last linked post is very relevant to why I think the standard counterexamples fail.)
* Fairness Requires Randomness: I confess I've just never understood the appeal of Taurek-style resistance to picking the best option (and associated complaints of "unfairness"). But for those who do, why not just see the actual distribution of identities as amounting to a kind of (metaphorical) divine lottery?
Have I missed any good ones? I know there are (bafflingly) many non-consequentialists out there, so please do help out by posting your favourite objection -- or, if it's covered above, explaining what in my responses you find unsatisfactory!
[P.S. See also, 'Why Consequentialism?', for reasons in favour of the view.]