This is not a problem at all, unless you were trying to have this player lose a race, which seems like a very narrow goal to have as a DM.

Worried about combat? Make some closed area encounters (caves, single rooms, etc) so that they can't take advantage of their super speed. Have things attack them on their way past. Ambush them. If they move a certain distance away from the fight, make them roll perception to make sure they're "on target" when they try to run back into the fight if they don't have an easy line of sight back to it. Have a flying enemy. Send Reverse Flash after him.

If a player wants to make a speedster, that's cool. It shouldn't break the game. This game has wizards who literally throw giant balls of fire and you're more worried about a guy who can...run away real fast? This is not something you need to hard stop the player on. And if the whole game starts to ONLY be about the fast player, start putting in challenges where moving fast doesn't solve the problem (which from games I've played/run, is about 99% of problems)

Consider this: Usain Bolt ran 300 feet in 9.69 seconds which roughly equates to 31 feet per second and about six times as fast as a typical elf's base speed. I would consider him to have at least a 17 Dexterity (feet coordination alone does not a juggler make) and at least one feat. That said, Bolt ran a straight line and no one there was trying to stick him with a sword, fry him with a lightning bolt or put a couple arrows in his hide. But poor Usain didn't have magic! So your player can move 1000 feet in 6 seconds? Does he have inertial dampeners? Has he ever seen a bike moving 11.36 mph hit a wall? Doesn't sound that fast does it? Others have expressed that this power gaming has some serious limitations regarding Ki, Haste, and racial abilities but I really see this as only a small part of what the character would be looking at.  That poor Tabaxi has worked his way into a situation that could end very badly just on the basics of experience. Usain trained for years to be able to endure 10 seconds at about a fifth of the speed this poor kitten who really didn't train at all....

Just a little food for thought for you to serve the player when he runs headlong into the arms of (insert BBEG here).

From the OP,  it doesn't sound like you have a player that has actually achieved this, you are just wondering how to get around something like this.

Remember that ultimately, no matter how hard players minmax their characters, it is an arms race the DM can always inevitably win if they want to, whether thats a trap, an apocalyptic scenario, an unbeatable enemy, etc. Regardless of how much players push their characters, you can always scale encounters to them.

If you want to limit their speed, add things to make them question whether or not they SHOULD use their speed in a given scenario. Traps, limiting visibility, physical (or magical) barriers, homebrew enemies, etc. can all be used to limit them. A lack of space to move in could also do the same thing. Sure, they can move fast, but if they are in a small cavern with no real priority targets that mobility doesn't matter as much. You can also use difficult terrain, climbing, swimming, acrobatics requirements to go through an area at that speed, and so on. Sure you can run 1,000 feet, but those enemies are 60 feet up a wall in a cave system. 

I think its also fair to at times let them have their power trip, and make encounters you KNOW they will dominate and look really cool in. 

Also remember that the character is limited by what they can PERCEIVE. How far can they see? 300'? Maybe? If that's the case, then when they declare their action to 'move to here' then they move to the point they can see and they stop. Seeing 1000' is hard under ideal conditions.

As with all things, if it becomes too much of a burden, sit the player down and talk it out. They should be reasonable enough to see that it's giving you grief and agree on a compromise.

I would also like to point out that the more a character makes themselves known in a battle, the more likely the enemies will be likely to target him/her.

Let him run around like crazy, but if he is really a thorn in the enemies side then absolutely they will single him out.  The rest  of the characters will have to work extra hard on crowd control.  Sure in a couple of encounters it could be a real bonus to have a super fast guy, but other times it could end up being a liability. 

If the player is strictly a min-maxer number cruncher then let them have it.  If they are a true RPer then have fun with it.  I would say as a DM, it depends a lot on the intentions of the player.  Are they trying to make a super cool character or just break the game?  

I'm all for DMs coming down hard on players that just want to break the game.  

Bruh just give some of your enemies the sentinel feat

Edit: here's why:

Whenever you hit a creature with an opportunity attack, its speed drops to 0 for the rest of the turn. This stops any movement they may have been taking.

• Creatures within your reach provoke opportunity attacks even if they took the Disengage action.

• When a creature within your reach makes an attack against a target other than you (and that target doesn't have this feat), you can use your reaction to make a melee weapon attack against the attacking creature.

I see no reason at all that a doubling doubles another doubling.   Just rule they work on the base.  So a 40 doubled and doubled and doubled = 80.

In other words, doubling does not stack.  And it should never stack.  Additions stack.

Quote from Mog_Dracov >>

I see no reason at all that a doubling doubles another doubling.   Just rule they work on the base.  So a 40 doubled and doubled and doubled = 80.

In other words, doubling does not stack.  And it should never stack.  Additions stack.

 RAW, they do in fact stack. 

In fact, if you give up a few levels of Monk for a few levels of Barbarian you can actually squeeze extra 5 feet into your base movement as well because of Fast Movement:

Fast Movement

Starting at 5th level, your speed increases by 10 feet while you aren’t wearing heavy armor.

A Monk 15/Barbarian 5 with Mobile feat ends up with base Movement of 75 feet. With Boots of Speed that’s 150 feet, and Haste that’s 300 feet, and Feline Agility that goes to 600 feet*4 (move, dash, BA dash, Haste Action Dash) that’s 2400 feet of movement in one round, or just shy of 273 mi/hr.

Of course, that was everything they could possibly do that turn, no attacking whatsoever. And after that burst they are down to half that speed on following turns (so 1,200 ft/round) until the spend a turn not moving and that is still everything that can do, no attacking whatsoever. 

don't really understand. a character's mobility has somehow broken combat encounters? opportunity attacks, ranged attacks, difficult terrain, spells that slow one down, aggro--what kind of monster likes some dipshit zipping around?--the list goes on. 

Simple: Break their legs, problem solved, your welcome.

My DM recently used fear to stop my advance. I don't have long range options and I kinda wrecked his boss fight previously by being able to move so much. It's fun in RP though when If I  chase someone. 

Movement and distances are a pain in the butt. Moving great distances isn't really a huge deal. You can always have the traps and sentinel and pa's people mentioned, but generally distance isn't a game breaker.

Theater of the mind and all... There's a way to play without maps. :P

Look, the general rule is that if you are in rage of something, then that something is generally in range of of you. 

Unless they're doing disengage actions, you should still be able to hit them. And if they are, there's the rest of the party to focus on...

"We'll, since you're out of range to attack, I guess we'll have to focus 20 goblins on the wizard and the cleric..."

Which is another point: that one character still has one action and bonus action. Action economy is going to kick that party's ass and with them staying.out.of range, the party's going to be pissed that he's conveniently never found around in a fight. 

The biggest problem though is toxic builds like this that are meant to cheat the game end up inviting more toxicity.l, so beware.

this is actually not completely correct. the better answer is it depends on the level of the character and stage of the game. 
a level 20 speed focused character can reach a max speed of 4960 ft in one round (6 seconds)
This is done with multiclassing and stuff. 
Tabaxi + haste + boots of speed. 
2 levels into fighter for action surge, 2 levels into wizard (bladesinger), 6 levels into barbarian, 10 levels into monk. finding a way to cast zephyr strike, using wizard spells to cast longstrider, having the boon of speed (which lets be real, by this point they had reached over 2000 ft in one round already, you know what they are after and what they want, grant them the boon and let them be happy, it really wont make much difference anyway), mobile feat, etc. 

this gets to over half the speed of sound, nearly letting them break the sound barrier by moving. and due to being monk, this includes across water or liquids and up vertical surfaces. 
The action surge, normal action, haste action, and bonus action are all used to dash. (zephyr strike can be cast using tricky ways not needing a bonus action, or if they do need to use a bonus action for it, it reduces their speed to around 3500 feet) (haste can be gained with a potion of swiftness)

and boom, you have a character zipping around faster than most living beings can reasonably react, once every other round or so (noticeably less after the first round due to the inability to keep casting zephyr strike)

They also have high ac due to having monk + fighting stance + bladesinger ac boost + haste 
so they are a bit hard to hit, which fits cause they are going so fast they genuinely would be hard to hit. 

But what is the issue here? they sacrifice a ton of damage by doing this. The only thing they can do to affect combat in a meaningful way is kiting the enemy or doing guerilla tactics. they have a relatively low health pool due to the monk and wizard levels. they give up most end game benefits from getting classes to over 10. and they give up most powerful spells due to only having level 1 spell slots. they are limited to 1 action surge. and they still do basic damage due to not having any truly game breaking things like rogue sneak attack. They also give up the ability to concentrate on things due to being in rage mode (though none of the spells they need require them to concentrate, only haste does but they are unable to cast it themselves)

so overall, their impact is honestly not all that great, they have some interesting utility for sure, and they certainly fit a very special role in the party. so let them. let them fit that very specific role in the party, they can be a great scout and that is good. if anything be grateful they are the scout, cause i promise you, a bugbear rogue assassin who went all in on getting as much bonus dice to roll is far more campaign breaking than this harmless speedster is.

i do not really see how it is a toxic build. it serves a purpose, and among the different things a character can min max, speed is mostly harmless. it is super easy to handle and super easy to also punish. but like at the end of the day, they are just fast. they give up a lot of ways to contribute meaningfully in combat. I would not say it is toxic, it is about equivalent to someone picking a bard. they do one thing and they do that one thing very well, but aside for that one thing, the rest of the things they try to do will only bring the party grief. tons of people still play bards though. speedsters are a bit rare. 

polearm master and narrow spaces with glaive users shut this build down in its tracks. as do traps, though be careful with those, you can rapidly lead into a player death situation very fast using certain traps.

As others have said sentinel, lots of enemies, npc holds an action then casts hold person, forcecage, banish, tight maps, environmental hazards, traps, encounters where the group is teleported in - I mean honestly I would let the player have their fun if they put all that effort into speed their dps sucks - if they need to kill the thing to move forward all the speed in the world does not kill the thing that needs killing. Honestly I am surprised their group mates don't call them out for a silly spec that is not a contributor to the overall party success. 

Or really just have a bunch of caster mobs teleport your melee mobs every turn - tons and tons of ways to end this. 

Last thought you could rule by fiat and just say movement is capped @ 150' at my table if you are looking to not be on the maps then I do not see why you should be part of the adventure as the adventure takes place on the maps. - A bit harsh and the least creative answer but its an answer. 

Depending on the condition of the battlefield, then you could say they take damage for moving that fast. At that speed, hitting a single grain of sand would be like a bullet. Say that if they only run, say, 500 feet, then they don't take damage.

1000’/ round is roughly 120 mph, illusion of open space hiding a wall or tree trunk - welcome to a splat, new character please ( or at least they are down and out for the rest of the fight. Don’t want a big something? How about a waist high stump with a broken limb at groin height? I once hit a car’s tail fin playing frisbee - ouch! Down for over a minute and that was only 10 mph. 

The thing is, it's really rare for battlefields to be more than a couple hundred feet across in the game just due to limitations on table space.  So this super max speed burst honestly is something that's more theorycrafting than functional.  Also, it requires on getting one or more magic items, which are not something that's automatically available to PCs at will.

I have a couple of players who have built speedy characters. One often does 180ft in a turn, and it takes resource to do it.

Honestly, from experience, I have yet to have it be anything but an awesome plotpoint. One time he did a quicksilfer-run around the room to pick up fallen items. Another time he tried to save several party members by distributing healing & handing potions out in one round.

If a character is built to move 1000ft in a round, then ok, they can. This may have them dart in and out (opponents quickly learn to prepare attacks, which are not opportunity attacks), or the fast character ends up overwhelmed with their party 940ft. behind them and running as fast as they can.

If you want to put the fear in them, have a monk opponent with stunning strike prepare an attack for when they enter range. dart in 500ft, and then stop moving. Oh dear, now it's the enemie's turn.

Honestly, I've had far more problem with ranged characters with long ranges than I've ever had with high movement speeds; a lot of the things that will deal with a sharpshooter plinking from 600' will also deal with extreme movement speeds.