One of the reasons you sometimes see a Time Spiral in Doomsday sideboards is because we can efficiently use it in Doomsday Piles for large amounts of storm. In my opinion these piles fill up a niche between the requirements of Chain of Vapor piles and that of double-Doomsday piles, which are both useful for generating high stormcounts. Time Spiral also comes with limitations, you will give your opponent a fresh 7 card hand.

The cost listed in square brackets [..] is assumed to be payed for by the lands that are untapped by Time Spiral. The listed stormcounts do not include for cards that may be cast for no other purpose then to build storm. In other words, cards like Conjurer's Bauble are assumed to not be recast post-Time Spiral. For most piles only the Lion's Eye Diamonds and Burning Wish into Tendrils of Agony are counted towards storm, in order to list the lowest possible post-Time Spiral mana cost.

A simple example of a pile that uses Time Spiral is as follows:

GP + 1 card (in hand or post-DD graveyard):
-> IU, LED, LED, BW, BW BBB+1UUR+[1R] (7) 11 Storm

If you have no other cards in hand or your post-Doomsday graveyard resolving Time Spiral will kill you. The Doomsday you've cast to build this pile typically ends up as the only card in your graveyard, with 5 cards in your pile this adds up to 6. However, the Burning Wish used to wish for Time Spiral exiles itself, so that's only 5 cards. That's is why you need at least 2 arbitrary cards in your hand or post-Doomsday graveyard to not kill yourself. You may also have more than 2 extra cards and be unable to get rid of them. This can be a problem as important cards may end up in your library rather than your hand after Time Spiral has resolved. This usually isn't a problem, unless your only draw spell is Brainstorm and the number of cards left in your library is less than 3 or when there are more cards left in your library than you reach with your draw spells, but these situations are quite rare. Given that you have at least 2 extra cards that you may cast before and after the Time Spiral, as long as they don't exile themselves or other cards, you might be able to up to storm to well beyond 11.

Also note that the listed cost assumes that you can produce 1R with the lands that you untap with Time Spiral. This is important because if you don't have any lands that produce red mana, and none of your 'extra' cards are Lotus Petal or Lion's Eye Diamond, you have to add R to the listed cost for the pile.

Executing a Doomsday-Time Spiral pile will become much easier if you have some resources that can produce red mana, like a Lion's Eye Diamond or Lotus Petal. Considering the following piles:

GP + LED:
-> IU, LED, LED, BW, BW BBB+UU+[1] (5) 13 Storm
-> IU, LED, LED, BW, BW BBB+R+[1R] (4) 13 Storm

These piles are also good examples of that you can sometimes pay 3 mana instead of 2 before Time Spiral in order to float a red mana and thus combo off without a land that produces red. However, if you that 3 additional mana that is any of 1UU or BBB you could also build a double doomsday pile.

Using additional cantrips for cheaper piles

Having access to a draw effect rather than a Lion's Eye Diamond is unfortunately less effective in Doomsday-Time Spiral piles. Consider the following (intentionally bad) example:

CB in play + GP:
-> IU, LED, LED, BW, BW BBB+2UU+[1R] (7) 11 Storm

These piles are very inefficient and in these situations it would be better to build a double-Doomsday pile, which costs BBB+UUU or BBB+BBB if you have SDT+GP.

It is not possible to turn additional draw effects into mana, because we don't have enough room in the pile and the whole procedure is also quite expensive mana-wise: IU=2 + BW=2 + TS=6 = 10 mana, with another 1R post-TS. As the 2 Burning Wishes and the Ideas Unbound already take up 3 spots, we only have room for 2 Lion's Eye Diamonds and these can be easily drawn by Ideas Unbound, extra draw is not going to help.

It's important to understand that the only way we can build cheaper piles is to cram more acceleration spells into the pile. Since IU and 2 BW already consume 3 slots, the only way we in which we can lower the mana cost of the pile is to replace any of these with LEDs. This is of course only possible if you already have Ideas Unbound or Burning Wish before casting Doomsday. The slot that is freed up will be used for an additional Lion's Eye Diamond. Examples:

CB in play + IU in hand:
-> LED, LED, LED, BW, BW BBB+UU+[1] (5) 12 Storm

GP + IU in hand:
-> LED, LED, LED, BW, BW BBB+UU+[1] (5) 13 Storm

Rather straightforward, cast IU to draw 3 LEDs that produce 9 mana. Then tap SDT or play GP to draw the Burning Wish sacrificing the LEDs in response. As Burning Wish + Time Spiral is only 8 mana, you can float R after casting Time Spiral and only need 1 more mana from your lands post-TSP to cast the Burning Wish for Tendrils of Agony.

Similarly, if you already have a Burning Wish before casting Doomsday, you might cram another LED into the pile:

2x GP + BW in hand or graveyard:
-> LED, IU, LED, LED, BW BBB+R+[1R] (4) 14 Storm
-> LED, IU, LED, LED, BW BBB+1R+[1] (5) 14 Storm

Now these last few piles use 3 extra cards rather than 2, meaning that you may likely have 8 cards in your library before Time Spiral draws 7. This could result in a situation where a crucial card, like Burning Wish, ends up in your library rather than your hand after Time Spiral. Be aware of the possibility, but this usually isn't a problem.

Using Brainstorm to draw into Doomsday-Time Spiral piles

Since we need some extra cards to not kill ourselves with Time Spiral anyways, we can use Brainstorm to efficiently draw multiple cards. Consider the following pile:

BS in hand + 2 cards:
-> P, LED, LED, BW, BW BBB+2UU+[1R] (7) 11 Storm

Now this pile is quite efficient, the mana cost is similar to having an Ideas Unbound+1 card in hand instead of Brainstorm, but we replaced Ideas Unbound in the pile with a Ponder that doesn't lower the mana cost. It's best to use a Ponder here to draw the Burning Wish that is the 3rd card in our pile. Although Brainstorm and Sensei's Divining Top (at -U+2 mana) also work. The advantage of Ponder over Brainstorm here is that if you need a card that is stuck in your post-Time Spiral library you can use Ponder to draw it, where using Brainstorm with less than 3 cards in your library would kill you.

Because the Time Spiral piles can only get more efficient if we can replace the IU or BWs in the pile, we seek to use Brainstorm as a draw3 and forgo the use of Ideas Unbound entirely. The problem is that in the above pile Ideas Unbound was replaced by a Ponder, which doesn't bring us any closer to lowering the cost of the pile. However if we already have a cantrip we can stuff another LED in the pile.

BS/Ponder + Brainstorm + 2 cards:
-> LED, LED, LED, BW, BW BBB+UU+[1] (5) 13 Storm

CB in play + Brainstorm + 2 cards:
-> LED, LED, LED, BW, BW BBB+U+[1R] (5) 12 Storm

And if you have either a BW or a cantrip, but less cards in total, you might as well use Brainstorm to end up with 2 LEDs and a cantrip, while Burning Wish is on top of your library.

BS + BW + GP:
-> LED, LED, BW, LP, LP BBB+2U+[1] (6) 13 Storm

BS + GP + 1 card:
-> LED, LED, BW, BW, LP BBB+2U+[1] (6) 12 Storm

+1 storm if you cast GP post-TSP. The Lotus Petals can be included to lower the post-TS cost of the pile, but when you have sufficient lands you may also include other cards like Silence or even Chain of Vapor.

Summary

I hope this gives a good overview of what Doomsday-Time Spiral piles are out there. Obviously, I have only touched upon the subject and the total number of piles that may be created is much larger. So things to remember are:

  1. You want at least 2 cards, beyond what you search with Doomsday
  2. You need 2 Burning Wishes, one for Time Spiral, one for Tendrils of Agony
  3. Additional cantrips can lower the manacost if you can replace either IU or some of the BWs in the pile with LEDs
  4. Instead of lowering the mana cost by adding LEDs you might as well include cards like Silence/Chain of Vapor/Slaughter Pact/Lotus Petal
  5. These examples intentially do not use TW or AoI. Both of these work badly with Time Spiral as they remove cards from the game that are needed to ensure your card count reaches seven post TS.