KidKevin:

I agree. Once carried a game with only a chalice and ionian boots. That time I was trolling though... had 5k monies at the end. But still carried with no ap items.

The main reason people say Veigar is a bad choice is because he is easily harassed early game. I fixed that with hp pots and teleport. They hit you, just heal it. And they never expect you to suddenly turn aggressive and drop the dark matter on their heads after 100 ap is farmed.

In follow up, I've developed a new build for Veigar, with the same requirements to use as the boots Veigar i.e. really good at last hitting.

I present: Attack Speed Veigar

Meki Pendant first, with two hp pots.
Build it into Nashors.
Boots of choice.
Wit's End
Malady
Madrad's Bloodrazor
Sword Of The Divine

The Nashors Tooth combined with Veigar's passive will be all the mana regeneration you'll really need. Q spam, last hit, etc. The order they are bought may vary based on the opposing team. Switch the Wit's End for a Phantom Dancer if you want.

After buying the Elixer of Agility, you'll have 2.5 attack speed.

You'll also have 300ish ap.

The damage ratio when hitting a turret is 0.4 (or something like that) for ap casters. I've found that each attack will do around 68 damage (or that could be Sword Of The Divine, but that's irrelevant). The turret will fall extremely fast. That is, of course, if you reach this far in the build.

In other news, you will still one-combo most casters (as long as you see them before they see you). Tanks will be little problem for you, due to Wits End and the Bloodrazor. Even a tanky dps, say Jarvan, I out damaged by auto attack. Open with the event horizon, dark matter, baleful strike combo as always, that'll take off 500-700 hp. It will also leave your target stunned a while longer. Right click to win.

This build has been tested twice with (relatively) good results (compared to the "Veigar likes shoes" build).