Not Updated For Current Season
This is just my basic Veigar build I can normally tap off around 1.1k AP with this build 1.4k without the Rylais and another Archangels staff. But it seems to me that the 1.4k AP without any extra health is not worth it. If you can mid as Veigar you can normally get a good 80-90 AP before having to back which your first buy will most likely be Sorc boots and a Tear. I personally like to not harass with Veigar until lvl 6 because you can catch most people (Ashe,Sivir,Kennen etc) off guard and kill them in a quick E,W,Q,R. I have done many other item builds on Veigar with Deathfire grasp, RoA and many other items but not seem to match the pure crazy AP of multipul Archangels since it is the highest giving AP item in the game when you have 3 of them.
Good luck with the build and tell me how it turns out.