The question may be aiming to expand one's scope beyond what's considered in introductory textbooks. This is the essence of scientific research.
The heat capacity $C_X$ at a condition of constant $X$ is $C_X\equiv T\left(\frac{\partial S}{\partial T}\right)_X$, with temperature $T$ and entropy $S$. We can interpret this as the heating required to obtain a certain temperature change (at constant $X$).
Introductory treatments often assume that a gas is enclosed in an an impermeable container (constant $N$); thus, we have the associated standard constant-volume and constant-pressure heat capacities
$$C_{V,N}\equiv T\left(\frac{\partial S}{\partial T}\right)_{V,N}\,\,\mathrm{and}$$
$$C_{P,N}\equiv T\left(\frac{\partial S}{\partial T}\right)_{P,N},$$
which for the ideal gas are conveniently equal to $\frac{dU}{dT}$ and $\frac{dH}{dT}$, respectively, with the internal energy $U$ and enthalpy $H$ potentials. (Note that no subscripts are required in this special case because they're irrelevant; for the ideal gas, the coefficients of $dV$ and $dP$ in the corresponding fundamental relations are zero.)
OK, enough review of introductory material. Consider now a gas at equilibrium with an adjacent material. (The system boundaries still include only the gas.) The corresponding heat capacities are now relevant:
$$C_{V,\mu}\equiv T\left(\frac{\partial S}{\partial T}\right)_{V,\mu}\,\,\mathrm{and}$$
$$C_{P,\mu}\equiv T\left(\frac{\partial S}{\partial T}\right)_{P,\mu},$$
with constant chemical potential $\mu$, as the number of gas molecules is no longer constant; the gas can diffuse into and/or react with the other material. (Note that this analysis requires care because the system entropy is now affected not just by heating but also by mass transfer, as mass carries its own entropy. One obtains an infinite heat capacity for some simple condensation models, for example, because no amount of cooling can lower the gas temperature while it's condensing—it just accelerates the condensation rate at the boiling temperature.)
Still broader, consider a magnetic gas; that is, a gas for which another type of work other than pressure–volume work is relevant: magnetic field–magnetization work, or $B$–$M$ work.
We can define new potentials
$$\Phi\equiv U+MB=TS-PV+\mu N+MB;$$
$$\Psi\equiv\Phi-MB,$$
where we distinguish $\Psi$ from $U$ by the possibility of $M$–$B$ work when the former is used. We might now heat a gas at constant magnetic field $B$ or constant magnetization $M$, respectively corresponding to heat capacities
$$C_{Y,B}\equiv T\left(\frac{\partial S}{\partial T}\right)_{Y,B}\,\,\mathrm{and}$$
$$C_{Y,M}\equiv T\left(\frac{\partial S}{\partial T}\right)_{Y,B},$$
where $Y$ refers to some combination of variables we already discussed ($V$, $P$, $N$, $\mu$) being held constant. Note that it can still be convenient to work in terms of a potential: $C_{M}=\frac{\partial \Phi}{\partial T}$ and $C_{B}=\frac{\partial \Psi}{\partial T}$, where the natural variables are being held constant. This is analogous to $C_V=\frac{\partial U}{\partial T}$ and $C_P=\frac{\partial H}{\partial T}$. (In other words, $\Phi$ and $\Psi$ are just as valid physical properties of the gas as the more familiar $U$ and $H$!) Recognizing such analogies and how they can be extended to an arbitrary degree is how one's thermodynamic muscles strengthen.
This gives eight possible heat capacities, with the possibility of extending the framework further with other work types. And we haven't even gotten into ways that functions of the thermodynamic variables might be held constant, rather than just the variables themselves. An infinite number of variations are possible, so the answer to the original question is (3).
