Why?
My comment makes sense, as the direction of time is thought of as increasing entropy: Time increases = Entropy increases.
So why does the OP assume time is infinite, but not entropy?
The OP is only a problem because it takes time as infinite and not entropy - “therefore over infinite time we ought to have reached maximum entropy infinitely long ago”… unless they were both finite, or both infinite - as the 2nd law of thermodynamics suggests, they track one another, and one may as well simply be the other. So they should both be assumed to be either infinite or finite, ridding the OP of its issue.
It’s interesting to think that there is a point at which “no work can be done and nothing will do anything” - suggesting maximum entropy and therefore maximum time: a final limit on both that is “the end of the universe”. So in that case, if they are infinite, that infinity stretches out behind that “end” unlimitedly (i.e. there is no beginning of the universe).
For both to be infinite, either there is a beginning and no end (no maximum entropy), there is an end but no beginning (no minimum entropy) - or neither beginning nor end, in which case entropy/time is increasing similarly to a “Shepherd Tone” - seemingly never ending or suggesting any beginning. Either that or it’s all finite with a beginning and end. Or multiverse etc. But whatever the case, the OP problem is invalid.
Well until someone comes up with a believable answer to that question, “What’s on the other side of that boundary to everything”, I’m going to have to go with an infinite universe. “Nothing” is not an answer.
Same issue with, “What was there before time began?”
I think the issue you’re encountering here is in the intuitive “everyday” conception of space and time, rather than thinking in terms of relativity where spacetime can curve. If you think about the curvature of spacetime reaching a maximum, that resembles a boundary without being one. And what curves spacetime to a maximum? Gravity, which is at a maximum when mass is at a maximum - say at a singularity where all the mass of the universe is compacted into a point like theorised by the big bang? This is also consistent with time dilation and length contraction being at a maximum closest to the speed of light, which is the kind of speed everything is travelling at with the electromagnetic force maximally overpowering the gravitational force in the closest of quarters, such as in a singularity. Shortest distances with length contraction? Check. Longest durations with time dilation? Also check. So time eases in from a maximum point, maximally slowly - simulating a start to time, without being a beginning. Not really infinite because otherwise it would never start, and not really finite because there is no “line” that presents your problem of “what’s on the other side”?
So basically I think your question has already been answered by relativity.