Turbo Fog
Simple enough, it tries to play Howling Mine, Temple Bell or Rites of Flourishing in either of its first three turns of the game, maybe alongside an Altar of the Brood.
This sets up additional card draw for all players. We use that card draw to end up drawing more than 1 Fog effect per turn of which we have 21 in the deck (so as soon as we draw 3 additional cards per turn, our odds become quite good).
From then on we use 1 Fog card per turn to stay alive. Dropping additional card draw effects to speed up the game. Altar of the Brood is only there to speed up the 'milling' process, it is not a win condition and is almost always sided out.
Our opponents deck themselves slightly before we do as they 'benefit' from our global card draws one upkeep before we do each time we play one, giving us a deck size advantage.
The Elixir of Immortality main board is used once around 10-12 of our lands are on the battlefield, we use it to shuffle the Fog effect cards back into the library, increasing their percentage compared to the start of the game for more consistent draws.
The deck is naturally weak against burn, so Dawn Charm and Riot Control have been included. A switch of a further 2 Elixir of Immortality to the mainboard would not be unreasonable. Pithing Needle is there mainly for maindeck plainswalkers but also has versatility against some rogue strategies with utility creatures.
Suggestions and questions very welcome. :)
When suggesting cards, please keep in mind the maximum price of this deck + sideboard must always remain under $40 on the price calculator as it is a budget endeavour.