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The etherealness spell changes the mechanism of movement so that you can move up or down at will (albeit slower than normal movement):

You step into the border regions of the Ethereal Plane, in the area where it overlaps with your current plane. You remain in the Border Ethereal for the duration or until you use your action to dismiss the spell. During this time, you can move in any direction. If you move up or down, every foot of movement costs an extra foot.

If I just keep moving upward will something ever stop me? Can I fly into the sun?

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Etherealness does not impose a limit on your height

Spells only do what they say they do, and nothing additional, pending DM fiat. If Etherealness doesn't specify a limit, like a height cap, then it doesn't have a limit.

If I just keep moving upward will something ever stop me? Can I fly into the sun?

The only limits to your new Etherealness-based Space Program are:

  • Your comparatively slow travel speed—at 15'/round, you're moving at 2.5'/second, which means to travel the distance of 490 Billion feet separating you from the sun, it would take (assuming 16 hours of travel and 8 hours of rest, requiring 3 7th-or-higher-level spell slots per day for Etherealness) 9,322 years to arrive at the sun. You can solve this problem by picking up Ancients-Paladin levels until you hit 15, and no longer age.
  • Any difficulties you might have with breathing in the near-vacuum of space—I recommend picking up a Necklace of Adaptation, an Uncommon Wondrous Item, which allows a user to breathe in any environment.

So if you're a multiclassed level 15Ancients-Paladin/13Sorcerer, and manage to acquire a Necklace of Adaptation, and occasionally spend a third-level spell-slot to conjure food + water, then you should be good to go with your nine-thousand-year-long suicide pact.

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    \$\begingroup\$ This answer almost makes me want to ask what the fastest way to fly into the sun would be... \$\endgroup\$ – David Coffron Jun 29 '18 at 18:34
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    \$\begingroup\$ Note that this assumes that you're playing in a world that has a concept of "space" and with its sun far away like Earth's. If you're playing in a world where the sun is a ball of pure magic that the gods move along the sky or somesuch, the specifics could be different. \$\endgroup\$ – user37158 Jun 30 '18 at 2:06
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You will not make it to the sun.

To begin, while in the Border Ethereal, visibility is limited to 60 feet (DMG p48). So it is entirely feasible to navigate through a city on the Prime Material Plane while in the Border Ethereal. But navigating through large unpopulated areas (i.e. space) accurately with no way to check bearings would be extremely difficult to nigh on impossible.

Etherealness also has a duration of 8 hours. If you are a spellcaster with access to 9th level spell slots, you could have up to 24 hours of Etherealness active before needing a long rest. Assuming you take the Dash Action and have a standard speed of 30, you move at 60 feet per round/600 feet per minute/3600 feet per hour/86,400 feet per day. As moving vertically costs an extra foot per foot moved, you're effective per day distance you can travel is 43,200 feet. Converting to miles, we get 8.16 miles per day. But that's combat speed. The fast travel pace from the DMG (p182) lists moving at a fast pace, you will cover 30 miles a day without exhaustion. Because of the difficulty in moving vertically in the Border Ethereal, let's say you can move 15 miles a day without risking exhaustion. The sun is about 92.96 million miles from Earth (according to the first result from Google). The exact distance doesn't really matter, as a. we're discussing a D&D approximation of Earth, and b. the order of magnitude is so large. So lets do some rounding. Say your sun is 90 million miles away and you can travel 100 miles a day. It's still going to take you 2,465.75 years to get there.

So aside from needing to be suicidal enough to want to undertake this venture, and having a way to circumvent the food requirements, you need to live for far longer than any of the standard playable races.

But Liches be crazy, right?

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    \$\begingroup\$ That last line... so dangerously on the nose with memes... \$\endgroup\$ – Slagmoth Jun 29 '18 at 19:01
  • \$\begingroup\$ I don't see any reason I couldn't recast an extended spell etherealness after a long rest. Also, the light from the sun (or other celestial bodies) would exist within 60 feet, so you could still navigate to it using star positions (sailors use precise measures of star positions to navigate the seas so this wouldn't be much different. There are also many ways to extend your lifetime and have higher movement speeds using magic. \$\endgroup\$ – David Coffron Jun 29 '18 at 19:08
  • \$\begingroup\$ At the risk of going off on a tangent, you can see light from an object without seeing the source object. This is common on a heavily overcast day. Even shadows can be difficult to read due to the scattering effect of the atmosphere. We have no idea what the environment of the ethereal plane does to scatter light, but since it reduces visibility to 60 feet, I would imagine that you would get no sharp shadows. This is deep into DM fiat territory, though. \$\endgroup\$ – keithcurtis Jun 30 '18 at 15:52

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