- a monument to the History of the Empire: a huge coil of granite slowly & constantly extrude from the ground, and an army of stone carvers surround the base, writing the events of the present on the thing as it emerges from the ground. An escalating spiral stair or floating disk-elevator goes up the center, and great balconies & terraced gardens protrude from the thing, like a vast 3-D garden that spiral upward thousands of feet. At the ever-rising pinnacle, a symbol of the emperor, a light, a watchtower, &/or the like, visible for dozens of miles.
Conceited person that I am, I don't like every idea in the thread, but I do like this idea alot. However, I think it works even better if, as it emerges from the ground it carves itself with the most important historical event in the Empire of that day. The reason I like this alot better is that its a much better campaign/adventure hook this way, as what happens when today's most significant historical event is something no one recognizes or which, although known, seems utterly trivial.
You can imagine the Roman emperor having one of these, and the days carving rolls up with a baby in a stable surrounded by donkeys and sheeps, and everyone going, "What the heck does that mean?"
Take that idea, adjust the event portrayed to suit your campaign. Also, this can create a really cool story reward for the players latter in the campaign, when the day rolls up and its something the players did that is on it.
- a Mobile Imperial Palace: either floating/levitating in air, driven by immense golden sails, arcane engines or flying beasts; OR on great enchanted wheels and pulled by a vast army of earth elementals, bulettes, giantkin, or the like; also with vast pontoons allowing it to float across the sea. In this way, the Emperor is always at home, no matter where in his realms he is.
Like it (postulated something similar), but this borders on 'too practical'.
"Census of All Things of the Empire, Both Great and Small" is commissioned, in order to very precisely determine exactly how extensive the holdings of the Empire. The results are tabulated and housed in a huge bureaucratic department, access to which allows virtually any question to be answered... eventually. Such an accounting of Everything, of course, can lead to great insecurity on the part of an emperor who really wants it all.
Again, this might just be 'too practical'. The same divination techniques that allow you to census everything in the empire, also would allow you to magically find the answer in the records. The only way this works is if cost/limits of magic really is a consideration and the census is in some way bankrupting, but given that magic can create and fabricate objects that may not be true.
"Department of Impossible Things Made Probable", for the more academically inclined Emperor, which is basically a research institute into creating paradoxes, untying Gordian Knots, and the like. End each problem created &/or solved is credited to the brilliance of His Imperial Majesty, naturally.
Again, this is too practical for my tastes. Having a whole department of miracle workers can be really really handy. The only way I think this idea works is if the solutions to all the problems make everything worse (think 'Bloody Stupid Johnson' from the Discworld books).
In the spirit of the original request, I think the suggestions ought to be things that are emmensely expensive but serve little or no practical purpose.