Where is the wiggle room in current gravity theories? As far as I know, General Relativity has long since been proved experimentally to every qualified person's entire satisfaction, and modern technology such as GPS relies on its accurate predictions.
So although there may be debatable aspects, such as local conservation of energy, and of course a theory of quantum gravity is still being sought, there seems little scope for improvement except possibly more streamlined formalisms and explicit solutions in more scenarios.
But skimming ArXiv articles, one often sees titles of papers which appear to refer to lots of variant gravity theories. So I was curious to know how these differ from GR.
Some look like abstract extensions to higher dimensions, perhaps with a view to elucidating why the 3+1 dimensions are favoured by nature. But others seem to require different laws of gravitional attraction, and one wonders if they are all consistent with GR. But, assuming they are, or are intended to be, where is the flexibility in GR which allows this variety?
 A: To understand where the "wiggle room" in general relativity is it is useful to look at one of the main theorems that constrains GR, Lovelock's theorem. This says that if we start from an action that


*

*is local

*depends only the spacetime metric

*is at most second order in derivatives of the metric
and

*is in 4 spacetime dimensions
then
the only possible equation of motion for the metric is the Einstein equation (possibly with cosmological constant).


The conditions of this theorem immediately tell us what assumptions we need to let go off to find alternative theories. You can


*

*Consider theories with more fields than just the metric, as is done for example in scalar-tensor theories such as Brans-Dicke

*Consider actions that contain higher derivatives of the metric. For examples in f(R) gravity.

*Consider non-local actions (I don't know of any good examples that people study.)

*Consider theories in different number of dimensions than 4.

A: I meant to add another answer to my question, but due to a mix up with user IDs I've been unable to log in until now!
Anyway, an interesting paper that covers exactly this topic was published a couple of weeks ago on the ArXiv: "An Introduction to Modified Gravity", by A Yu Petrov
https://arxiv.org/abs/2004.12758
"This paper is a collection of lecture notes on modified gravity. Various modified gravity models formulated within the Riemannian formalism are discussed."
A: I do not see any "wriggle room" in general relativity. The mathematical structure can be derived from only one axiom, together with a couple of elementary empirical facts. The axiom is

The general principle of relativity: Local laws of physics are the same
  irrespective of the reference matter which a particular observer uses to
  quantify them

which itself can be reduced to 

Hume’s principle of uniformity in nature: the fundamental behaviour
  of matter is always and everywhere the same.

As Hume observed, without this principle, there would be no science. If fundamental behaviours could change, nothing could be concluded from observation

“For all inferences from experience suppose, as their foundation, that the future
  will resemble the past, and that similar powers will be conjoined with similar
  sensible qualities. If there be any suspicion that the course of nature may change, and that the past may be no rule for the future, all experience becomes useless,
  and can give rise to no inference or conclusion.” — David Hume, 1748, An
  Enquiry Concerning Human Understanding, Sec II, Part II.

The general principle is combined with Einstein's argument for special relativity, that space time coordinates are what we measure and are defined by the way in which we measure them, and with local conservation of energy and momentum as expressed in Einstein's equation for gravity. The Einstein tensor is the unique rank-2 tensor formed by contraction of the Riemann curvature tensor,
together with the metric, which is symmetrical, and which satisfies 
the contracted Bianchi identity, which means that Einstein's equation summarises the two principle tenets of Einstein’s theory of gravity, that matter is the cause of curvature, and, through the contracted Bianchi identity, that energy-momentum is conserved and hence that inertial objects follow geodesics. 
Although people may try relaxing Einstein's equation, even if it is possible, it seems that doing so could only produced results in conflict with these fundamental tenets. Local energy-momentum is a fundamental requirement of particle interactions. I see no more other wriggle room than the cosmological constant. Indeed, if one takes the argument for special relativity to its logical conclusion, then I do not even see room for the cosmological constant; coordinates are determined by local measurements, not by boundary conditions at infinity (I have argued in The effects of turbulence generated in Big Bang nucleosynthesis that the cosmological constant is not needed for a match between general relativity and observation; see also refs therein).
Incidentally, there is no issue over local conservation of energy, which is contained in Einstein's equation. The issue arises because (a generalisation of) potential energy can be stored non-locally in geometrical structure. I don't have any problem with this.
The only wriggle room concerns situations in which it is not meaningful to define spacetime coordinates by measurement processes. This includes singularities; more importantly, it is precisely the situation described in quantum mechanics. I have argued that general relativity and quantum mechanics are in fact already unified on a fundamental philosophical and mathematical level by a relationist approach which take this into account in my books and in Mathematical_Implications_of_Relationism.
It remains a question as to why nature should prefer 3+1 dimensions (at the moment my best guess is that it is the least number of dimensions which works), but I really do not hold with theories which try to modify gravity with other forces, or which seek to replace the empirically established structure of curved spacetime with an interactive force (e.g propagated by gravitons) on an unobservable and empirically meaningless flat spacetime.
Before one looks at ways to modify general relativity one should make a thorough study of the principles on which is is based. One good place to start  Newton's argument for Absolute Space and Absolute Time in the scholium to the Principia, since he stated with complete clarity (and indeed much more clarity than anyone until Einstein) that only the relative concepts are observable, from which we should conclude that only the relative concepts have place in science. 

“I do not define time, space, place, and motion, as being well known to all. Only I must observe, that the common people conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.” — Isaac Newton, 1689, Scholium to the Definitions in Philosophiae Naturalis Principia Mathematica

As Leibniz remarked, as Einstein showed in special relativity, and as von Neumann showed in quantum mechanics, the proper way to formulate science uses only the relative quantities.
