According to our current understanding of physics, there are several factors that would prevent a spaceship with unlimited fuel from continuously accelerating to near the speed of light.
Firstly, as an object with mass approaches the speed of light, its relativistic mass increases, requiring a larger amount of energy to accelerate it further. This means that the closer you get to the speed of light, the more energy you need to add to achieve the same increment in speed. As you approach the speed of light, the energy required to accelerate further becomes practically infinite, making it impossible to reach or exceed the speed of light.
Secondly, as you approach the speed of light, the effects of time dilation and length contraction become significant. Time dilation means that time appears to pass more slowly for a moving object relative to a stationary observer. Length contraction means that the length of an object in the direction of motion appears shorter to a stationary observer. These effects become more pronounced as you approach the speed of light, making it increasingly difficult to achieve further acceleration.
Furthermore, there are practical limitations in engineering and materials. As you approach relativistic speeds, the energy and stresses involved become immense, requiring technologies and materials that are currently beyond our capabilities. It's not just a matter of fuel, but also the propulsion systems and structural integrity of the spacecraft that would need to be considered.
In summary, due to the increasing energy requirements, time dilation, length contraction, and technological limitations, it is currently believed that it is not possible to continuously accelerate a massive object to or beyond the speed of light, even with unlimited fuel. The speed of light acts as a fundamental speed limit in our universe according to our current understanding of physics.