Lossless Transmission Line Length vs Rise Time
What you're describing is a known, old problem. You are using the ideal translmission line (tline), which should output no overshoot, no matter what, yet they clearly exist. This has nothing to do with the tline, but with the way LTspice, and SPICE, in general, work: the timestep is variable and it is halvened, or doubled, depending on the speed of the simulation, which is dictated by the rates of change (the derivatives) -- if it's smooth, the simulatino flies, if not, it has to reduce the timestep to calculate the sharp(er) variations. If these are too sharp (i.e. discontinuities), the solver may not be able to halven the timestep anymore and it will throw  "timestep too small" errors.

In this case, the circuit is simple, with only one sharp rise, which propagates. This means that the solver would normally "fly" through the simulation, but it has to break around the edges. And here, the default, compressed waveform display comes into play: it is by default, compressed to only 300 points, and thus it has to connect the dots somehow. This, coupled with the relative speed of the simple circuit, causes some points to be miscalculated, and the errors propagate. If you'd prolonge the chain of transmission lines, you'd see more and more distortion.

The solution: force a timestep. Simply add .tran 0 2.5n 0 1p. This forces LTspice to no longer fly through the rest of the circuit more than the imposed timestep, which causes tighter tolerances, thus more accurate results.

It's always good to remember that, no matter the simulator, they are all limited by the machine precision and, usually, they use float unless forced to double. In addition, the tolerances (abstol, reltol, etc) are even more relaxed, in order to not force you to wait eons for simulating a simple transmission line propagation test.