First hard Theoretical Limit:
One limit, even in space with no atmosphere, with almost infinite mirrors.. is the Sun’s temperature. When it reaches the temperature of the Sun, it must be emitting as much radiation at the Sun as the Sun is emitting at it. So its temperature cannot increase further. (The other question’s replies say only that, as only that was asked. Not a duplicate. E.g. no enquiry about...
Number of Mirrors
Assuming we’ll have a sufficient temperature for radiation to dominate convection, the number of mirrors needed grows quickly:
If $\alpha$ is its absorptivity, $N$ is the number of mirrors, $w$ is the power in $1m^2$ of direct sunlight, $r$ is the fraction of light perpendicularly incident (or equivalent thereof), then the total power absorbed is: $$P_{in}=\alpha r wN$$
Power plant near Primm, NV where they point a couple hundred thousand huge, curved mirrors at a water tank and make it white hot:
https://en.m.wikipedia.org/wiki/Ivanpah_Solar_Power_Facility
To maximize temperature, we wait for equilibrium, $P_{in}=P_{out}$. Scanning a solar power site, $w=600 \frac{W}{m^2}$ is a guesstimate for an earth summer temperate-zone. Achieve $r=$~$0.5$. For emissivity of $\epsilon$, and the Stefan-Boltzman constant $\sigma$, power balance is $$N\alpha 300=\epsilon \sigma A T^4$$
By the Kirchoff Theorem, at that point (equilibrium), $\alpha = \epsilon$. Yes good emitters are usually good absorbers, and exactly so at equilibrium.
The previous equation (and common sense) says to minimize surface area, but still catch the whole beam from our $1 m^2$ mirrors. OP made a good call with a $1m$-edged cube, $A=6m^2$. $~ \sigma= 5.67\times 10^{-8}$ in SI (mks). Plugging those in:
$$N \approx 10^{-9}~ T^4$$
It’s interesting that a poor absorber/emitter reaches the same temperature as a good one, as both directions of the power balance are affected. However, if we worried about getting up to a $T$ where convection is meaningless, then we’d want high $\alpha ,\epsilon$.
Assuming it hasn’t melted, some points on this curve in (deg-C, $N$ mirrors) are:
$(200, 49) ~(500, 350)~ (1000, 2600) ~(2000, 26.5k) ~(3000,114k) ~(4000, 330k)$
It may seem like $330,000m^2$ of flat mirroring is implausible, until we see what’s been done (well over a million $m^2$ of curved, focused mirroring, see image).
The biggest immediate benefit I can see would be concentrating the incoming light on one side and somehow getting higher absorptivity/emissivity there with a composite or varying surface treatment. If we could get $0.8$ there and $0.2$ on the other five sides, then the effective absorptivity would be $0.8$, but the effective (net) emissivity would be between $\frac{0.8+5(0.2)}{6}=0.3$ and $0.8$ - considering the object’s temperature gradients. If it was $0.5$ for example, then $N$ would go down $38$%.
Any practical limits?
Absent unmentioned constraints, it depends almost entirely on how exacting the mirror specs can be with respect to their shape and positioning (an engineering question so not considered; curving the mirrors would help a lot). Another salient factor is the object’s altitude. For a ground location, there would be less total heat at equal precision.
Adding more sunlight will always add more heat, but the net heat levels off and cannot reach sun temperature through earth’s atmosphere. That’s because once it’s hot enough, the object will be transmitting heat mostly via radiation, at a rate proportional to $T^4$. Even in space, it eventually emits back as much as it takes in. Otherwise we would violate the Second Law of Thermodynamics with heat flowing from something cold to something hot in a reversible process.
