I heard it was 100 feet above minimums, I'd just like verification.
From the Pilot/Controller Glossary:
CLEARED TO LAND−ATC authorization for an aircraft to land. It is predicated on known traffic and known physical airport conditions
A clearance to land is exactly that: a clearance to land. A pilot without landing clearance hasn't violated ATC instructions until their aircraft has, in fact, landed. So the answer to how close you can get is: anything that isn't touching the surface.
Imagine going for the "I'll come within 1 foot then go-around" option at night when due to a comms failure a plane was lined-up (also, ATC is allowed to withhold the landing clearance in this scenario), or there is a departing plane that has just rejected its takeoff and the comms were blocked due to a simultaneous transmission.
While nothing seems to prohibit that option, it isn't the best of advice. While not regulation, AC 91-73B is aimed at flight schools, and advises to go-around if landing clearance is not received and could not be confirmed when on final approach (no height is specified).
Instruct students that on final approach, if they have not received landing clearance, to ask the tower, "Flight School call sign, am I cleared to land?" and, if there is no response, to execute a go-around.
A pilot that is trained with that mindset, should keep it after finishing the flight school – always ask, or as the AIM puts it:
ATC will normally withhold landing clearance to arrival aircraft when another aircraft is in position and holding on the runway.
i. Never land on a runway that is occupied by another aircraft, even if a landing clearance was issued. Do not hesitate to ask [emphasis added] the controller about the traffic on the runway and be prepared to execute a go-around.
In short, there is no minimum, but best practices and good judgement (not hesitating to ask) should apply.
On a related note:
Did You Know?
There have occurred collisions and incursions involving aircraft holding in position awaiting a takeoff clearance. The FAA's analyses indicate that two minutes or more elapsed between the time a line up and wait instruction was issued and the resulting incident. CURRENT GUIDANCE IS TO CONTACT ATC AFTER HOLDING IN POSITION FOR 90 SECONDS.
(Runway Safety, FAA)
TL;DR there is no minimum distance
An aircraft landing on a runway of a controlled aerodrome may do so for a variety of reasons.
For example they might be on final approach and due to communication loss they don't get a final clearance and land anyway.
In a controlled aerodrome you are usually given a clearance to enter the TMA (terminal movement area) as well as a STAR (standard arrival route). Entering a TMA without clearance will surely cause more trouble, especially at busy and / or sensitive aerodromes.
There are even incidents, where pilots landed on the wrong runway or even the taxiway. Also landing at the wrong airport in the same TMA has happened.
Landing on uncontrolled aerodromes is also regulated. There is usually a frequency to listen to and a pattern to fly, to make sure you are the only one landing.