Is there a maximum energy for a relativistic particle?

There is no maximum energy of a freely moving massive particle in special relativity.

The relativistic energy of a particle of rest mass $m$ moving in your frame at speed $v$ is given by $E=\gamma m c^2$ where $\gamma = \frac{1}{\sqrt{1-(v/c)^2}}$. If you look closely at $\gamma$ you will see that it is not defined at $v=c$ ($c$ is the speed of light), and that $\underset{v\rightarrow c}{\lim}\gamma = \infty$. From this you conclude that the energy of a particle increases without bounds as its speed approaches that of light.

From this you see that: 1) you can always increase the energy of a massive particle by accelerating it, 2) you need more and more energy to approach $c$, so no massive particle can travel at the speed of light.

It's helpful someday to learn a correct expression for the energy of a particle. An example is $$E=\sqrt{(mc^2)^2+(\vec pc)^2}.$$

Which works for any mass (even zero) and any momentum (assuming the particle is on shell, which classical particles are). But it requires you to know the momentum. And the momentum of a massive particle at the speed of light, like the energy, is infinite.

You should expect momentum to increase without bound becasue a force isn't mass times acceleration. A force is actually (even to Newton) the time rate of change of momentum. So apply a force and the momentum changes, keep applying and the momentum keeps increasing. It's just that momentum isn't equal to $m\vec v.$ Instead $$\vec p=\vec v E/c^2,$$ or $$p=\sqrt{(E/c)^2-(mc)^2}.$$

And you might wonder why I didn't write an equation for $p$ in terms of mass and velocity. And the reason is because you can't determine the momentum from the mass and velocity alone. It simply isn't a function of mass and velocity. For a massive particle you can write $$\vec p=\frac{m\vec v}{\sqrt{1-\frac{v^2}{c^2}}},$$ but this isn't a fundamental relation because it only holds for massive particles. So fundamentally, momentum isn't determined by mass and velocity.

Momentum and energy are the actual fundamental things. Mass just tells you how they balance together through $$(mc^2)^2=E^2-(\vec pc)^2,$$ and velocity is $$\vec v=c^2\vec p/E.$$

A passive variant of Andrea Di Biagio answer is to consider a particle at the supposedly maximum energy and then to consider evaluating that energy in another reference frame that moves in the opposite direction of that particle.