I’m late to this, but I’d like to bring up something I haven’t seen anyone else mention. But first, some more details regarding what has been discussed:

In most situations, it’s correct to say that EM waves basically don’t interact with one another. You can cross two laser beams, and they’ll just continue on their way without caring that the other one was present. A mathematically equivalent scenario is waves on a string: the propagation of a wave isn’t affected by the propagation of another, even when they overlap. Another way to put this is that they obey the principle of superposition: the total amplitude at any given point on the string is just the sum of the amplitudes of the individual waves at that point. You may want to argue that the waves do interact because there are interference effects, but interference is exactly what you get when they don’t interact, i.e. when the principle of superposition holds.

However, this is only true for so-called linear systems. I won’t go delve too deep into the math of what this means, but I think looking at the wave on a string example can give you some intuition. The behavior of waves on a string can be explained mathematically by treating the string as a large number of tiny points connected by springs. If the force on a given point by a neighboring spring is directly proportional (i.e. linear) to the spring displacement (Hooke’s law), then you find that the entire system obeys the wave equation, which is a linear equation. This is the idealized model of a string, and the principle of superposition holds for it perfectly. If, however, the forces acting on points within the string have a non-linear dependence on displacement, then the equation describing the overall motion of the string will be non-linear and the principle of superposition will no longer hold perfectly. In such a case, two propagating waves could interact with one another as the properties of the wave medium (the “stretchiness” of the string) would be influenced by the presence of a wave. In other words, the stretchiness of the string would change depending on how much it’s stretched (e.g. if a wave is propagating on it), and the stretchiness influences the propagation of waves.

Something analogous can happen with EM waves, and has been mentioned by others. In so-called non-linear media, the electromagnetic wave equation becomes non-linear and two beams of light (propagating EM waves) can influence one another through the medium. This makes sense when you consider that the optical properties of a material can be changed, even just temporarily*, when enough light is passed through it (for example, by influencing the state of the electrons in the material). It makes sense then that this modification to the optical properties of the material would influence the propagation of other waves through it. In the string example, this is analogous to the string itself being modified by the presence of a wave (even just temporarily) and thereby influencing the propagation of other waves. Such effects require sufficiently large wave amplitudes to be noticeable, i.e. the intensity of the light needs to be high enough to appreciably modify the medium.

What about the case of light propagating in a vacuum? If the vacuum itself is the medium, surely it can’t be altered and no non-linear effects could arise, right? In classical electromagnetism (Maxwell’s equations), this is true. But within quantum electrodynamics (QED), it is possible for the vacuum itself to become non-linear when the strength of the electromagnetic field is great enough. This is known as the Schwinger limit, and reaching it requires extremely high field strengths, orders of magnitude higher than what we can currently achieve with any laser.

*I want to emphasize that we’re not necessarily talking about permanent changes to the medium. In the case of waves on a string for example, the string doesn’t need to be stretched to the point of permanent deformation; non-permanent changes to its stretchiness are sufficient.