Return to Answer

Forced March. The Travel Pace table assumes that characters travel for 8 hours in day. They can push on beyond that limit, at the risk of exhaustion.

 

For each additional hour of travel beyond 8 hours, the characters cover the distance shown in the Hour column for their pace, and each character must make a Constitution saving throw at the end of the hour. The DC is 10 + 1 for each hour past 8 hours. On a failed saving throw, a character suffers one level of exhaustion (see appendix A).

Forced March. The Travel Pace table assumes that characters travel for 8 hours in day. They can push on beyond that limit, at the risk of exhaustion.

For each additional hour of travel beyond 8 hours, the characters cover the distance shown in the Hour column for their pace, and each character must make a Constitution saving throw at the end of the hour. The DC is 10 + 1 for each hour past 8 hours. On a failed saving throw, a character suffers one level of exhaustion (see appendix A).

This could be achieved with the following level 20 character. Level 10 Moon Druid, Level 5 Barbarian, Level 2 Monk, Level 2 Rogue

This could be achieved with the following level 20 character. Level 10 Moon Druid (with the Mobile Feat), Level 5 Barbarian, Level 2 Monk, Level 2 Rogue

Level 10 Moon Druid, Level 5 Barbarian, Level 2 Monk, Level 2 Rogue

Either 360 or 270 feet, depending on what you mean by "not-time limited"

All movement is limited by time somehow. Note the rules on movement over long distances in the Players Handbook (PHB, p. 181):

Forced March. The Travel Pace table assumes that characters travel for 8 hours in day. They can push on beyond that limit, at the risk of exhaustion.

For each additional hour of travel beyond 8 hours, the characters cover the distance shown in the Hour column for their pace, and each character must make a Constitution saving throw at the end of the hour. The DC is 10 + 1 for each hour past 8 hours. On a failed saving throw, a character suffers one level of exhaustion (see appendix A).

So for example, if your character was to attempt to travel for 24 hours straight, they would need to make a 16 Constitution saving throws (with DCs of 11 through 26, increasing by 1 each hour). The probability of making all of these saving throws is very low (about one in 323 assuming you have a +11 on each save), and once you fail two of them your speed halves (see "exhaustion" PHB, p. 291).

This consideration makes me conclude that you may only be interested in travel methods that can be done for up to 8 hours in a day. However, if this is not the case and you want ones which can be done for more than 8 hours (in spite of the perils of exhaustion), I've included an "unlimited" option (or you could just see Nautarch's excellent answer.)

If you only need 8 hours in a day (in 5 hour increments with a one hour break in between): 360 feet per turn

This could be achieved with the following level 20 character. Level 10 Moon Druid, Level 5 Barbarian, Level 2 Monk, Level 2 Rogue

1. Level 10 Moon Druid uses two uses of Wild Shape to transform in to an Air Elemental (MM. p. 125).

Fly speed of 90 feet.

2. Level 5 Barbarian gives Fast Movement, and Level 2 Monk gives Unarmored Movement. +10 movement each.

Fly speed of 110 feet

3. Take the Mobile Feat. +10 feet of speed

Fly speed of 120 feet

3. Dash every round

240 feet of flying movement per turn

4. Use the Rogue's Cunning Action to Dash again each round with a bonus action.

360 feet of flying movement per turn.

The Druid will be able to remain Wild Shaped for 5 hours (half their Druid level) before reverting into their normal form. At that point, they can take a short rest (1 hour) and transform back for another 5 hours. This would mean they could easily travel for 8 hours in the day.

If you don't want to be limited by 8 hours of travel, 270 feet per turn

You could use the exact same build described above, but instead transform into an Eagle. This would result in a flying speed 30 feet slower than the Air elemental, meaning a total of 90 feet less movement per round (270 per turn). Unlike the Air elemental, this method would permit you to fly for about 10 hours straight by expending one use of wild shape every 5 hours. If you took a short rest (which you'd probably need at least to eat), you'd then be able to fly for another 10 hours straight.

However, if you want to have the full capabilities of your class while transformed (as opposed to those actions only performable by an Eagle), you could use Nautarch's build and have the same movement speed while still being a completely capable Monk/Rogue.