RAW there is no specified limit, however, the general rules of movement and travel apply with their risks.
You have not specified a setting with established rules, hence this will be general. Your DM will narrate the world and they will think of such things as "Is there such a place as space", "When does space start?", all things that make for the rules of that world are established and narrated by your DM. In 5e we have no tables that define how long you can see things in detail, only that you can even discern fine detail up to a mile when you have The Aspect of the Beast: Eagle (PHB 50).
You will have to ask your DM how high you can fly and drop objects with accuracy or even see creatures for that matter. You may be able to see as far as 40 miles, but perhaps there is always fog or clouds in the sky that will obscure your vision.
Visibility Outdoors (DMG 243):
When traveling outdoors, characters can see about 2 miles in any
direction on a clear day, or until the point where trees, hills, or
other obstructions block their view. Rain normally cuts maximum
visibility down to 1 mile, and fog can cut it down to between 100 and
300 feet. On a clear day, the characters can see 40 miles if they are
atop a mountain or a tall hill, or are otherwise able to look down on
the area around them from a height.
The sky is a place with encounters like any other place, but you will be able to stay out of range of most creatures.
The Sky (DMG 119-120):
Flying characters can move from one place to another in a relatively
straight line, ignoring terrain and monsters that can't fly or that
lack ranged attacks.
Flying by spell or magic item works the same as travel on foot, as described in the Player's Handbook.
As adventurers travel through the air, check for random encounters as
you normally would. lgnore any result that indicates a non-flying
monster, unless the characters are flying close enough to the ground
to be targeted by non-flying creatures making ranged attacks.
Characters have normal chances to spot creatures on the ground and can
decide whether to engage them.
So you may face encounters on your way towards space if there is such a thing in your DMs world. And you have the normal travelling pace, see Travel Pace (PHB 181). And as you are a first-level Aarakocra I will assumme that you have no gear or means to avoid Exhaustion (PHB 181, 185, 291), so you will eventually fall to your death. That is if your DM didn't come up with hazards such as thin-air, air-currents, etc. that prevented you from flying as far up as you want.
Or perhaps your DM enjoys your idea and will provide you with a little hut in the clouds, that has endless support of rocks to drop on your enemies, while you sit in your stronghold and giggle at their demise.
You will have to get the things that you drop on your enemies somewhere because you have a limited carrying capacity, but this is one of the reasons why the ability to fly is considered very powerful. You can solve a lot of encounters easily, especially if you are playing one on one.
The other hazard that you will eventually face is High Altitude (DMG 110):
Traveling at altitudes of 10,000 feet or higher above sea level is
taxing for a creature that needs to breathe, because of the reduced
amount of oxygen in the air. Each hour such a creature spends
traveling at high altitude counts as 2 hours for the purpose of
determining how long that creature can travel. Breathing creatures can
become acclimated to a high altitude by spending 30 days or more at
this elevation. Breathing creatures can't become acclimated to
elevations above 20,000 feet unless they are native to such
environments.
High altitude will severely impact the upward-progress that you make and gives insight into some of the expected boundaries by game-design. Perhaps you want your tiny hut with an endless supply of rocks here between 10,000-20,000 feet, where you still can get acclimated without being native to the area.
