The real limit is not based on the size or number of particles, but on the coherence of the group of particles. Using the word coherence is probably not helpful without context, so let me give a quick explanation of what that means.
As I mentioned in my answer above, particles can exhibit wave-like properties. A group of particles will each have their own wave packet. In our everyday lives, two particles, even those right next to each other, are jiggling around randomly due to temperature and experiencing slightly different environments. You can think of each of these separate random jiggles as a measurement that collapses the wave function of that particle. Then, after the particle's wave function collapses, it begins to evolve again until the next measurement. As a side note, saying a measurement collapses the wave function is quantum mechanics talk for the observed reality that when we measure where a particle is, we do not find a wave, we find a particle. So, the shorthand for this view of quantum mechanics is that a measurement collapses the wave function.
Ok, so now we have a bunch of particles, like a chair. Why won't a chair tunnel through a wall? Well, all the particles that make up the chair are not just physically separated, but they are jiggling due to their temperature and their slightly different environments. So, all of these particles that make up the chair keep having their wave function collapsed randomly. The chance of one particle tunneling through the wall is small. The chance of all 10^27 particles in the chair independently tunneling through the barrier at once is not going to happen before the universe ends.
Back to coherence. All of these particles of the chair are independently jiggling around, and each one has its own wave function collapsed very quickly. For this reason, you can treat each particle as an independent particle. We would say the wave functions of these particles are not coherent with each other.
Now, imagine that we have two particles right next to each other. At room temperature, they are constantly jiggling and having their wave functions collapsed. If we cool them down to reduce the jiggling, and they are close enough to each other, their respective wave functions can start to overlap. When the jiggling of the particles is small enough and their wave functions overlap sufficiently, they begin to behave as a single quantum entity. This is a coherent state. In a suitably constructed experiment, these coherent particles can then exhibit quantum behaviors such as tunneling together.
Back to your question: is there some maximum size for a group of particles that could be forced to maintain the correct state to pass through the wall? Since the group of particles must be in a coherent quantum state to tunnel, the real question is how big of a group can be put into such a state. You have to cool them to slow the jiggling, isolate them from anything in the environment that might collapse their wave functions, and get them close enough together for their wave functions to overlap. There is likely a theoretical limit that could be calculated, but as a practical matter, extraordinary engineering efforts are required to get even a very small group of particles into a coherent quantum state. The direct answer to your question is that while there may be a theoretical maximum possible size of a coherent state for our universe, the real limit is set by the immense practical challenges of creating and maintaining a coherent state. This is what makes the work of this year’s Nobel Prize winners so impressive.
The real limit is not based on the size or number of particles, but on the coherence of the group of particles. Using the word coherence is probably not helpful without context, so let me give a quick explanation of what that means.
As I mentioned in my answer above, particles can exhibit wave-like properties. A group of particles will each have their own wave packet. In our everyday lives, two particles, even those right next to each other, are jiggling around randomly due to temperature and experiencing slightly different environments. You can think of each of these separate random jiggles as a measurement that collapses the wave function of that particle. Then, after the particle's wave function collapses, it begins to evolve again until the next measurement. As a side note, saying a measurement collapses the wave function is quantum mechanics talk for the observed reality that when we measure where a particle is, we do not find a wave, we find a particle. So, the shorthand for this view of quantum mechanics is that a measurement collapses the wave function.
Ok, so now we have a bunch of particles, like a chair. Why won't a chair tunnel through a wall? Well, all the particles that make up the chair are not just physically separated, but they are jiggling due to their temperature and their slightly different environments. So, all of these particles that make up the chair keep having their wave function collapsed randomly. The chance of one particle tunneling through the wall is small. The chance of all 10^27 particles in the chair independently tunneling through the barrier at once is not going to happen before the universe ends.
Back to coherence. All of these particles of the chair are independently jiggling around, and each one has its own wave function collapsed very quickly. For this reason, you can treat each particle as an independent particle. We would say the wave functions of these particles are not coherent with each other.
Now, imagine that we have two particles right next to each other. At room temperature, they are constantly jiggling and having their wave functions collapsed. If we cool them down to reduce the jiggling, and they are close enough to each other, their respective wave functions can start to overlap. When the jiggling of the particles is small enough and their wave functions overlap sufficiently, they begin to behave as a single quantum entity. This is a coherent state. In a suitably constructed experiment, these coherent particles can then exhibit quantum behaviors such as tunneling together.
Back to your question: is there some maximum size for a group of particles that could be forced to maintain the correct state to pass through the wall? Since the group of particles must be in a coherent quantum state to tunnel, the real question is how big of a group can be put into such a state. You have to cool them to slow the jiggling, isolate them from anything in the environment that might collapse their wave functions, and get them close enough together for their wave functions to overlap. There is likely a theoretical limit that could be calculated, but as a practical matter, extraordinary engineering efforts are required to get even a very small group of particles into a coherent quantum state. The direct answer to your question is that while there may be a theoretical maximum possible size of a coherent state for our universe, the real limit is set by the immense practical challenges of creating and maintaining a coherent state. This is what makes the work of this year’s Nobel Prize winners so impressive.