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Well, first of all, I am not a physics major, just a high school student. Watching a bit of cosmos and some physics at school got me thinking about this.

Well, the earth exerts acceleration due to gravity of magnitude $9.8\;\mathrm{m/s^2}$ which causes deceleration in everything that we throw up, correct?

And if we assume light to be made of particles (photons), it should theoretically exert some acceleration on them as well, but I presume it would be very minimal so as to not cause any significant change.

But if we imagine an imaginary celestial body (and every celestial body has a finite gravity field, correct? beyond which gravity cannot act and henceforth exerting no acceleration towards it on any body) that has the gravitational force strong enough, so that it exerts a very strong acceleration on everything. If it's also luminous, a ray of light will escape from its surface at $3\times 10^8 \;\mathrm{m/s}$.

Now this time, imagine the acceleration due to gravity being so strong that by the time the light has reached the end of the gravitational field of that celestial body, the speed of light (or the photons, which should theoretically get slowed down) is now, let's say $20\;\mathrm{m/s}$.

The light, after escaping the gravitational field, considering there's no neighboring celestial body to exert a gravitational force on it, will maintain its speed of $20\;\mathrm{m/s}$. We can then send a satellite into space that can easily hit that speed, therefore traveling faster than what is still technically light.

Also, when we say it's impossible to travel faster than the speed of light, do we refer to a numerical value of $3 \times 10^8\;\mathrm{m/s}$ or the speed of photons? Because we can theoretically travel faster than photons? And if it's the numerical value, what are the basis to which we arrived at the cosmic speed limit?

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    $\begingroup$ Are you asking whether it's possible to catch up to a speeding photon? If you try to do that, the gravity that's retarding the photon will be retarding you too, so you won't be able to. You'll actually never measure that photon's speed as anything less than $c$ anyway - the effect of forces is a bit different when you get up to the speed of light. You might want to read this question and its answers $\endgroup$
    – Dawood ibn Kareem
    Commented Jul 5, 2017 at 11:50
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    $\begingroup$ Possible duplicate of Could gravity accelerate light? $\endgroup$
    – Hritik Narayan
    Commented Jul 5, 2017 at 12:17
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    $\begingroup$ You can't just compare speeds of objects that are in different gravitational environments, that's meaningless. In vacuum it is impossible to overtake light, that's essentially the basis for the theory of relativity and since this theory works so well and we have no evidence to the contrary, we assume it is a law of nature. However, it is possible for particles to travel faster than light in a medium. See for example the Cherenkov effect. $\endgroup$
    – Wouter
    Commented Jul 5, 2017 at 12:40

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so, when we say it's impossible to travel faster than the speed of light, do we refer to a numerical value of $3×10^8$ m/s or the speed of photons? Because we can theoretically travel faster than photons? And if it's the numerical value, what are the basis to which we arrived at the cosmic speed limit.

The speed of light hasn't been explained by any theory. We can only measure it and take it as a given, in the the same way that the mass and charge of an electron are based on measurements only. These are called free parameters, and there are about twenty of them, for which there is currently no accept underlying explanation.

We can only travel faster than light in fiction. In vacuum,the speed of light is the figure you quote above.

Please read this Free Parameters to follow the importance of free parameters.

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