This is an excellent question. You are talking about photons and other massless particles. In fact gluons are said to travel at the same speed c. Just to make sure you understand, photons and other massless particles travel at speed c in vacuum when measured locally. Gluons are always confined. Gravitons are said to be massless too, but they are hypothetical.
Now you are thinking this the wrong way. You think you can speed up to the speed c and you think normal is the spatial speeds that we (who have rest mass) experience.
Now after the big bang, there was a period of sea of photons. There was nothing else in the universe just photons. All were traveling at speed c, in vacuum when measured locally.
Now these photons were and still are moving in the time dimension with speed 0. they are not experiencing time as we do (who have rest mass). To experience time, as we do, you need to gain rest mass, and that will slow you down in the spatial dimensions and at that moment you will start moving in the time dimension, and you will start experiencing time as we do.
The reason for that is because the universe is built up so and the four vector is built up so, its magnitude needs to be c. Photons move in the spatial dimensions at speed c in vacuum when measured locally, so their speed in the time dimension is 0. Particles with rest mass travel in the spatial dimensions with speed less then c, so the speed in the time dimension has to compensate because the magnitude of the four vector needs to be c. So particles with rest mass start to move in the time dimension and start experiencing time as we do.
So speed c is not the maximum speed, it is the only speed, that photons in the early universe traveled at. Later when there was SSB and baryon asymmetry, particles with rest mass moved at speed less then c. Because to move slower then speed c you need to gain rest mass.
Now your question is why is the speed of massless particles speed c, in vacuum when measured locally and not an infinite speed. This is because the universe is built up so. There is no ether that photons would travel in. So it is not because of the limitations of the medium, because there is no medium. The real answer is we do not know. The universe is built up so, that even energy, that is photons need time to propagate. But that is only from our viewpoint. There is no rest frame of a photon. You cannot say if you would travel with the photon you would see this and that. But what we can say is that photons do experience time differently, and that they have a lightlike worldline, where the spacetime distance between emission and absorption is 0. The photon is experiencing it all in one. So you see, from the photon's viewpoint (there is no such thing) it might seem all instantaneous or as you said their speed could be seen as infinite. But here is no such rest frame of the photon.
It is only from our viewpoint (us who have rest mass and experience time the way we do) that we see the photon travel from emission to absorption in a certain nonzero time period. That period and distance in spacetime is 0 (lightlike). But from our viewpoint it takes some time, and some distance, and we as per the math divide them and get a speed, that is c. Again, that speed is c because we calculate it from our (having rest mass) viewpoint.
As the comments say too, the universe is built up so and anything else would brake casuality. Now it is like a misconception to ask, ok, but why does the photon not travel instantaneously? But if you ask that, then it should be obvious that you do not consider the lightlike spacetime interval of that travel, which is already 0. How much more instantaneous do you want? If you would see the photon travel instantaneously, then what would be the lightlike worldline, the spacetime distance of that? It is not possible for the photon to travel instantaneously from our viewpoint because it is already traveling 0 spacetime distance from emission to absoption, and you cannot travel shorter then that.