What is a clock? Relationalists love to define time operationally as what clocks measure, but this begs the question of what counts as a clock. Clearly, it's a measuring instrument and what it measures is supposedly time. We need to avoid circular definitions of time. There are many possible measuring devices, and most don't measure time. Which of them counts as a clock? To which one has to answer it has to give a linear real number as a pointer reading, it orders events correlationally, etc etc has all the properties we demand of time. This is circular.
 A: It's a good foundational question. There is the ordering of time (before and after) and the measure of time (how long after). I won't talk about ordering, since we all have a direct pre-theoretical sense of it. (You can see eg. Wikipedia's "arrow of time" article.)
As far as measure goes, time is what is measured by periodic processes. A periodic process is one that cycles in a fixed amount of time. This is circular, of course. 
However, there is a broad class of cyclical processes that are standard in the sense that they all proceed in very nearly a fixed ratio with respect to each other. On the longer scale we have astronomical cycles. On shorter scales there is also the swing of a pendulum, the vibrations of a tuning fork, and oscillations of the electromagnetic field (the current SI definition of the second uses the electromagnetic radiation from the hyperfine transition of cesium). These relations depend only on cycles (order of time) and on counting (whole number). Therefore this class can underpin time measurement without circularity.
A: 
[...] the question of what counts as a clock.

In order to define the notion of "clock" in general, in the context of (experimental) physics, it is instructive first to note certain more specific types of "clock" (whose explicit operational definitions may be more familiar already), namely 


*

*good clock (cmp. MTW Figure 1.9) as a participant "$A$" (given as ordered set of his/her/its indications: $A \equiv \{ A_{\alpha} \}$) with a real-valued parametrization $\tau : A \rightarrow {\mathbb R}$ such that
$ \forall A_{\alpha}, A_{\beta}, A_{\lambda} \in A \, {\rm with} \, A_{\alpha} \prec A_{\beta} \prec A_{\lambda}:$ 
$$  (\tau[ \, A_{\beta} \, ] - \tau[ \, A_{\alpha} \, ]) {\rm Duration}[ \, A_{\beta}, \, A_{\lambda} \, ] == (\tau[ \, A_{\lambda} \, ] - \tau[ \, A_{\beta} \, ]) {\rm Duration}[ \, A_{\alpha}, \, A_{\beta} \, ];$$ and

*ideal clock (cmp. MTW Box 16.4) as a participant "$A$" (given as ordered set of his/her/its indications: $A \equiv \{ A_{\alpha} \}$) of which there is a subset of "ticks" $T \subseteq A$ identified such that 
(1) each "tick" can be indexed by an integer:
  $\exists f: T \rightarrow {\mathbb Z}$, where 


*

*the indexing of "ticks" is distinctive:
$\forall A_{\alpha}, A_{\beta} \in T \, :$ 
 $$(f[ \, A_{\alpha} \, ] == f[ \, A_{\beta} \, ]) \iff (A_{\alpha} \equiv A_{\beta})$$,

*the indexing of "ticks" by integers is (strictly) monotonous to their order:
$\forall A_{\alpha}, A_{\beta}, A_{\lambda} \in T \, :$ 
 $$(A_{\alpha} \prec A_{\beta} \prec A_{\lambda}) \iff (f[ \, A_{\alpha} \, ] - f[ \, A_{\beta} \, ]) * (f[ \, A_{\beta} \, ] - f[ \, A_{\lambda} \, ]) > 0$$,

*the indexing is without gaps:
$\forall A_{\alpha}, A_{\lambda} \in T \,:$ 
 $$\forall k \in {\mathbb Z} | (f[ \, A_{\alpha} \, ] - k) * (k - f[ \, A_{\lambda} \, ]) > 0 \implies \exists A_{\beta} \in T | f[ \, A_{\beta} \, ] = k$$; and
(2) the indexing of "ticks" by integers constitutes a "good clock":
$ \forall A_{\alpha}, A_{\beta}, A_{\lambda} \in T \, {\rm with} \, A_{\alpha} \prec A_{\beta} \prec A_{\lambda} \,:$
$$  (f[ \, A_{\beta} \, ] - f[ \, A_{\alpha} \, ]) {\rm Duration}[ \, A_{\beta}, \, A_{\lambda} \, ] == (f[ \, A_{\lambda} \, ] - f[ \, A_{\beta} \, ]) {\rm Duration}[ \, A_{\alpha}, \, A_{\beta} \, ].$$
Prerequisits are of course the notions of "distinguishability" and "ordering" of the indications of any suitable participant (for which there may not be given any further definition, but which are instead rather a prerequisit in asking for any definition); and the notion of "duration" (or at least: "duration" ratios) for which the presciption of MTW Box 16.4 may itself be considered an operational definition (provided the necessary construction of "Schild's ladder" and identification of "free falling particles" could be followed without requiring the notion or even values of "duration" ratios in turn already).
Consequently, any "clock" in general, which is characterized neither as "ideal" nor "good", would have to be defined by even weaker requirements; suitably perhaps as a participant "$A$" (given as ordered set of his/her/its indications: $A \equiv \{ A_{\alpha} \}$) with a real-valued parametrization $t : A \rightarrow {\mathbb R}$ such that $t$ is a (strictly) monotonous function:
$$\forall A_{\alpha}, A_{\beta}, A_{\lambda} \in A | (A_{\alpha} \prec A_{\beta} \prec A_{\lambda}) \iff (t[ \, A_{\alpha} \, ] - t[ \, A_{\beta} \, ]) * (t[ \, A_{\beta} \, ] - t[ \, A_{\lambda} \, ]) > 0$$.
Examples of such general clocks are the various "forensic clocks", cmp. 
https://www.google.de/search?q=%22forensic+clock%22 which are not characterized by any particular "ticks" or "regularities", and which are not "good" by definition (but which may be found only incidentally more or less "good" by comparison with a "good clock" or an "ideal clock"; i.e. by measuring their "goodness" a.k.a. "accuracy").

Clearly, it's a measuring instrument

Clearly, of any given clock (except an "ideal clock" according to the prescription of MTW Box 16.4, as far as it can be followed at all) it can be asked and measured, whether it was "good" (a.k.a. "accurate"), or how it deviated from having been so, in any particular trail.
Therefore the defining and primary measuring instruments or setups are only any "ideal clocks"; while all other "clocks" would at best depend on a "calibration chain" and/or be subject to corresponding assumptions. 
A: A clock is a device that measures regular (repeatedly equal) intervals of time for an observer in the rest frame of the clock. There is no call for it to produce a pointer reading, and it most certainly does not order events.
A: Perhaps the question is better answered by asking, "what makes for a BAD clock". Most people know, at least intuitively, the answer to this question: a bad clock is one that shows (1) no direct correspondence to other clocks and (2) variations in the period after many cycles compared to another clock of the same type that has undergone 1 period. 
