You can seriously tick off a datacentre by putting an UPS inside your own rack.
What are the risks to the data center should a customer (somehow) choose to do this?
It interrupts the flow of the Emergency Power Off (EPO) in the datacenter. If there is a life or death emergency in a datacenter, that EPO might be triggered to save someone's life. If your rack has its own UPS, it will violate that EPO order.
You will not get extended runtime. Chances are as soon as your upstream UPS switches into battery mode, your UPS will detect the change in sine wave and drop into backup mode as well.
You'll violate your warranty and potentially your datacenter's UPS warranty. UPSs are warrantied to be installed in very specific scenarios and power sources. Your UPS-on-UPS is not going to be a supported configuration.
(from rexkogitans) You have to care about your UPS while there is a staff caring for a UPS that may be provided to you nonetheless. So it adds administrating a UPS to your work really unnecessarily.
Apart from low-probability scenarios, the single most important reason an unnecessary UPS really cheeses people off is that a year or two from now, it will start beeping.
An UPS needs regular battery replacements, and it usually tells about it by making a really annoying beeping noise that goes on forever. In a locked-up rack. Right next to the other rack you are trying to get some work done on. And it will keep making that noise for days (or months), because whoever installed it and didn't maintain it, also couldn't be bothered to configure any remote alerts on it.
And adding insult to injury, all this happened, because someone wouldn't trust the much better UPS system already in place. Yes, use an UPS when you have to place your rack in a utility closet of an 18th century farm house. But don't do it, when you have way more reliable power readily available on the outer side of the UPS.
You may need to run your own UPS when your datacenter doesn't quite deserve the name and it does not provide centralised UPS for you and other customers.
On the other hand, if that is the case there will probably not be a rule preventing you from installing your own though.
If you only get a couple of units in the top of a cabinet, good luck hoisting your UPS that high and not tipping over the cabinet, those things can be heavy!
And maybe stating the obvious: an implied consequence when you need to run your own UPS, is that everything that is not connected to it will not be protected. That may include your connectivity and other services the datacenter provides.
Don't forget the Electrical Code. Most have something like NEC 110.3b, "equipment must be used according to its labeling and instructions".*
That gives the instructions the same force of law as the Code itself.
If your UPS says "do not feed from a UPS" or if the datacenter's UPS says "do not supply UPS's", then you place the datacenter in dutch with the electrical codes. You don't get written up - they do.
* Because that is the scope of testing that was done when it was UL listed. This section is partner to 110.2, which requires only approved equipment be used. All this is on the first page of NEC. You can't miss it.
Most rack-style UPS units have, according to their label specifications, significantly higher input wattages than the equipment they are used to feed. No matter what alternate calculations or measurements or precautions YOU took, a third party planner or inspector HAS to assume that the power feed to such a unit needs to be dimensioned for what is on the label of that unit.
Also, while you might say "worst it can do, should it unexpectedly draw all of its rated power, is trip an overcurrent device, which is rated for the ampacity of all wiring behind it" - in a multi-tenant situation, this happening can affect other customer equipment. And every unscheduled reboot or disturbance of customer equipment, even if it is still within SLA tolerances, drastically worsens that customer's satisfaction with the provider.
TL;DR for next two paragraphs: Corrosive liquids, flammable gasses, electrocution hazards.
Also, all common battery systems are a chemical hazard - outgassing (lead acid batteries can generate flammable gas if a charger fails in the wrong way), leakage and subsequent corrosion (in the worst case of someone else's equipment. Lead-Acid batteries contain sulfuric acid, which is the very definition of a corrosive chemical!), fire (especially with new-school lithium systems that seem to be entering the market - also, a fire safety system designed to deal with an electrical or battery gas fire might not be designed to deal with a metal fire!)... if you bring in equipment of a brand/model/design and/or in a condition that differs from what the personnel in that datacenter knows well in practice, it will mean an unknown risk and extra training requirements.
Also, most UPS units can easily form an "IT" type mains on their own by accident - which presents an electrical hazard. This has nothing to do with IT, I am referring to "isole terre", meaning a floating-ground AC mains supply that could cause severe injury without any RCD type device interfering!
Personnel that is well experienced and trained in both types of hazards with their own choice of equipment could have a hard time dealing with your equipment if there is any incendent - not knowing eg how to shut it down NOW and RELIABLY if there is an electrical accident downstream of the unit, or how to quickly and safely remove a battery that failed catastrophically.
I can think of two things that this might help with, and that is providing you with the ability to network control your own power to nodes for the purpose of on-demand full node shutdown/poweron and power-based fencing for clustering.
Don't do it, though. If your colo does not offer these features and you need them, go elsewhere with your business.