It is important to understand that the speed of light is not a limit the way you think of it. The speed of light is c when measured locally, in vacuum. It is the same speed for EM waves and gravitational waves.
It is a common misconception to think that you can speed up to the speed of light. That is not correct. SR tells us three things:
anything with rest mass travels with a speed less then c, and will always do so
anything with no rest mass travels with the speed of light, and will always do so
anything that travels faster then the speed of light will always do so.
People usually do not even know about 3.
It is not that the speed of light is the maximum speed limit. It is just that anything that has rest mass will always travel slower then light (in vacuum when measured locally).
Now you are thinking about this the wrong way. You think you can speed up, but it is actually the other way around. The only speed is the speed of light. Light does not move in the time dimension, its speed in the time dimension is 0. Light does not experience time the way we do. If you want to experience time, you need to slow down in the spatial dimensions. The way to do that is to gain rest mass. If you gain rest mass, you will start moving in the time dimension, and start experiencing time the way we do.
In your case, your spaceship has rest mass. Let's assume it has unlimited fuel. As it starts to speed up, at about 0.8c, relativistic effects will become dominant, and ass the spaceship tries to speed up, it gains more mass. Having more rest mass will make it even harder for it to speed up. At this point, as it nears the speed of light, it would need infinite energy to speed up, so it becomes impossible.
Now there is an easy way to think about this. Anything in the universe has a speed relative to the speed of light (in vacuum when measured locally). Your spaceship has a speed of let's say 0.8c. When you do the slingshots, you just increase the speed to 0.81c etc. The increments in the speed of the spaceship are also relative to the speed of light. Even the lightest thing, the neutrino is traveling at a speed relative to the speed of light. As you go closer to the speed of light, the relative increases will need even more energy, and after a while, they will need infinite energy, so your spaceship will not speed up anymore, but will just gain more mass.
This form of use of relativistic mass is not in use in particle physics anymore, because it becomes confusing. In particle physics, they use rest mass, invariant mass.
I am using this example only for your question with the spaceship.
And I would like to mention a nice example for you. The way you describe it, imagine a neutrino traveling through space from far away, lets say passing by black holes, starts etc. Every time it passes by a star or black hole, it might do a slingshot, and speed up. By the time it gets to Earth, it would be possible to find a neutrino that has done slingshots the way you describe it and has sped up to speeds more then c. But we have not found any so far, and that is because even if the neutrino would do slingshots and speed up, it would still just speed up relative to the speed of light, and would always still travel slower then light.