DBWI: No Great Roman Wall

How would today's world be different if Emperor Trajan hadn't ordered to build a wall on the eastern European border of the Roman Empire?
I doubt it would be that different. Bauvaria would not have one of its bigger tourist attractions, but that's it.

Maybe Rome could have lasted past the third century without one of the initially better emperors losing their mind though.
Maybe there would be a weaker sense of national identity in Europe. And the barbarians from the east would have an easier time entering Roman territory.
While on the actual military side of things 'national border walls' for nations larger than city states are of limited value (and they would continue to be so until the advent of the modern nation state), the wall itself helps to focus local and regional economic development, and the construction and manning of them poured money and resources into those regions (remember a lot of retired soldiers were given land near the wall and settled there), and the 'gates' and garrison towns help to consolidate trade routes and migration patterns of many tribes.

So without the wall I'd hazard that the those regions would have been less developed, and a lot of borders would have been very different (as quite a few of the successor states have co-opted parts of the walls as a 'natural' border of sorts, the same way that rivers had often been convient borders).