Is fission/fusion to iron the most efficient way to convert mass to energy? Is fission/fusion of any element to iron-56 (or nickel-62?) the best way to convert mass to energy, that doesn't involve black holes?
In other words, will we be always limited to convert only about 1% of the mass available to energy? Are there other ways (using strangelets? antimatter?) to go beyond that limit?
I exclude black holes as, as I understand, you can only extract a finite amount of energy by reducing their spin, so they are not viable for energy production on a cosmological scale.
 A: You exclude black holes, but Hawking radiation is guaranteed to eventually radiate 100% of the mass energy you throw in as black body radiation. The temperature of the radiation is inversely proportional to black hole size. But if you could get your hands on a very small black hole (which would probably not look black at all), you could theoretically throw anything you don't want in and harvest the radiation.
One problem with this black-hole power plant is that as the size dwarves, the power output approaches infinity. That means that you MUST feed it, or evacuate it.
A: You exclude black holes, particularly the Penrose process for extracting the rotational energy of a spinning black hole. Forgive me for answering with black holes anyway. Penrose describes three methods in his classic review article "Gravitational collapse" (1969). These are completely impractical by the standards of human technology, but fun, and informative about physics concepts :)


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*Take pairs of black holes in space, let them spiral them around one another until they merge, and harvest the energy of the emitted gravitational waves. Form the new black holes into pairs and repeat the process. Repeat this enough and it approaches 100% efficiency of mass-energy conversion. This proposal is credited to Misner.

*Slowly lower an object on a rope towards a black hole. Use the force to turn a turbine or something. Think of the energy as released from gravitational potential (see Buzz's answer). Release the object at the horizon. This has up to 100% efficiency. Penrose claims doing this with a rotating black hole, one can release >100%, but without doing the calculations I am skeptical.

*The "Penrose process".
I have personally researched proposal number 2. Note Gibbons (1972) was the first to analyse the tension in the rope, although there is an error as pointed out by Unruh & Wald (1982) and Redmount (1984). We assume the rope is ridiculously strong. I have generalised from a quasi-static case to a moving rope in any static spherically symmetric spacetime, see MacLaurin 2019, "Cosmic cable", forthcoming hopefully in the proceedings of the 2018 Marcel Grossmann conference.
A: Friction in the accretion disk that forms a quasar, releases an enormous amount of energy, converting anything up to a third (~ 30%) of mass to.energy.
While this is powered by gravitational potential energy (massive black hole at centre), the energy is not actually released as matter meets antimatter, or at the event horizon as matter passes into the BH, or by Hawkings radiation, but by ordinary - and immense - friction.
So potentially, an accretion  disk round a neutron star, or other dense object, could have similar effects. NS would be the most obvious non BH candidate....
A: We don't know any better ways today, but we don't know everything.
The problem is that, as far as we know, the total baryon number is conserved.  This puts strict limits on what reactions are possible.
If you look at the linked page in the section "Conservation" you will see a couple of theoretical ways this conservation law might be breakable.  Obviously, we don't know how to do that, yet, but who knows what the future will bring?
A: Matter-antimatter annihilation, such as an electron annihilating with a positron to form two high-energy photons, can convert 100% of the mass into radiation. So fission and fusion are far from the most efficient ways to convert mass into other forms of energy. Unfortunately, the universe appears to contain almost no antimatter.
A: The most efficient non-gravitational way of extracting energy from ordinary matter is indeed to convert it into elements in the $^{56}$Fe region.  There is a fairly broad plateau of nuclides with binding energies of about $8.7$ MeV per nucleon, so it does not matter very much which of these you actually turn the source matter into.  (However, $^{56}$Fe is the optimum, since it has the lowest mass energy per nucleon, and the number of nucleons is conserved.  $^{62}$Ni has a lower binding energy per nucleon, but it is less stable, since some of its neutrons may be converted to protons to release additional energy; effectively the binding energy does not count the amount of energy that is tied up in the slightly greater neutron mass.  On the other hand, much of the energy in that would be released in the conversion of Ni or Co nuclei to Fe is in the form of neutrinos, which would be extremely hard to capture.)
However, the amount of energy that can be released via fusion still pales in comparison with what can be obtained using a deep gravitational potential.  This does not need to be a black hole.  A neutron star will do.  The release of energy in a core collapse supernova comes from the enormous gravitational potential energy that is unlocked when a $\sim1.4\,M_{\odot}$ white dwarf (with a radius of thousands of km) collapses into a neutron star (with a radius a just a few km).  If you already have a neutron star, you can drop matter onto it and capture the radiation it emits as it accelerates toward the neutron star surface.  Under optimal conditions, this can actually release a substantial [i.e. ${\cal O}(1)$—of order 1] fraction of a dropped particle's entire mass energy.
Finally, there is also direct matter/antimatter annihilation, which lets you get all the mass energy out of the source material.  However, it requires you to have ready sources of both matter and antimatter, which is not necessarily possible on large scales.  (You cannot manufacture the antimatter without putting in just as much energy as you intend to obtain from the annihilation reaction.)  So this option will not function with a generic hunk of source matter as your potential fuel.
