Word for: "Repeatedly doing something you are scared of, in order to overcome that fear in time"

One possibility is habituate:

ha·bit·u·ate

verb

make or become accustomed or used to something.
"she had habituated the chimps to humans"

From the NIH

The term “habituation” in the field of stress neurobiology refers to the reduction in physiological responses elicited by an nth exposure to a repeated homotypic (same) stressor in comparison to the large responses elicited by acute exposure to that stressor.

Another is desensitize: From the NIH

de·sen·si·tize

verb

make less sensitive.
"creams to desensitize the skin at the site of the injection"
    make (someone) less likely to feel shock or distress at scenes of cruelty, violence, or suffering by overexposure to such images.

I suggest acclimatize. Lexico has

acclimatize (British acclimatise)
_VERB

1 Become accustomed to a new climate or new conditions; adjust.
Getting acclimatized to life in Glasgow was tough at first; even after all these years away he still gets lonely and homesick.

Although not a complete answer, the phrase "successive approximations" is part of a process that includes desensitization, as an earlier poster suggested.

For example, in your dog illustration, to overcome a fear of dogs, the phobic person could simply look at some pictures of dogs, first in books and magazines, then in a friend's smartphone, then near a dog park from 50 feet away, then in the same general area but 25 feet away, and so on.

Each increment involves both a successive approximation and its corresponding--one hopes!--desensitization.

A similar procedure could be used in helping a person overcome the fear of elevators. First, have the phobic person observe from a distance people getting on and off the elevator, then move the person closer to the elevator to become more familiar with its features (e.g., the up and down buttons, the way people sometimes keep the doors open by blocking the closing sensor to allow a last-minute rider to get on, and the progress the lift is making, floor by floor by looking at the floor indicator lights).

Another successive approximation would be for the phobic person to walk into the elevator and immediately exit the elevator. A next step might be to go from floor number 1 to floor number 2--a short trip indeed! Then try going up 2 floors, then 3, then 4, and so on.

This technique of overcoming phobias, in psychology, is referred to as "desensitisation".

Note that the technique itself is "desensitisation" (a single word, as requested). The two-word "systematic desensitisation" specifically describes doing this desensitisation in a systematic way (e.g. starting small and increasing, rather than doing it arbitrarily or randomly), thus making it appropriate for use in clinical practice.

This wikipedia article gives an almost identical example, but with snakes:

A client may approach a therapist due to their great phobia of snakes. This is how the therapist would help the client using the three steps of systematic desensitization:
[Wikipedia]

One part of the CBT treatment process that's often used to treat simple phobias involves gradual exposure to your fear, so you feel less anxious about it. This is known as desensitisation or exposure therapy.
[NHS UK]

Less formally, get used to or accustomed to or comfortable with it.

Stress inoculation

This is a common term among military personnel. Consider how Navy SEALs are subjected to increasing levels of difficulty to prepare them for the most challenging scenarios (that would otherwise grip them with fear, crippling their ability to make decisions under pressure.) A typical progression of stress inoculation to escape from capture while underwater may resemble:

• Swimming
• Swimming while blindfolded
• Swimming while blindfolded and hands bound
• Swimming while blindfolded and hands bound and instructor randomly rips your oxygen mouth-piece out of your mouth to simulate unpredictable, real-world adversity

Flooding is a form of desensitization therapy.

It tends to focus on giving the patient a long, drawn-out exposure to the object of their fears, to give the panic time to wear off, rather than just doing it lots of times.

https://en.wikipedia.org/wiki/Flooding_(psychology)

An alternative might be conditioned:

From https://dictionary.cambridge.org/dictionary/english/condition :

to train or influence a person or animal mentally so that they do or expect a particular thing without thinking about it

That scenario could be described as the parents conditioning their child to be tolerant of dogs. This implies that the parents reinforce the child's calm behaviour around dogs through e.g. physical and emotional reassurance.

Another perhaps slightly-less-common word for this would be inure:

Definition of inure
^{transitive verb}

to accustom to accept something undesirable
// children inured to violence

Here, to add to the excellent answers already in place, is a candidate for the name of the action of the one who undergoes such a change.

Normalize:

to return to the usual or generally accepted situation (Cambridge)

When you normalize a situation or when it normalizes, it becomes normal. (Collins)

That is, in this case, the thing which he or she fears remains, but her or his perception of the feared thing changes:

Here it is in the wild, in a piece saying that Americans by and large have "accepted as normal" something that at one time would have seemed too terrible to consider as a possibility.

Over the last few months, election-preoccupied Americans have normalized what was once an unthinkable, and certainly an unconscionable, level of death and suffering. There have been a thousand deaths from the coronavirus a day, roughly speaking... (New York Intelligencer)

v. destigmatize

I think this thread would also be remiss without mentioning two common idioms: "to get over (something)", and to "get back on the horse/bike" (referring to getting back on after you fall off, to purposefully get over the stigma).

That being said, I think lots of the answers in here already are good, to condition one's self against..., to inure one's self (against), to inoculate against, desensitize.

To expose oneself to the source of a fear in order to overcome it is to face or confront that fear. I'm surprised that no answer or comment has mentioned this well-worn terminology.

In the context of mental health treatment, this is called Exposure Therapy. From wikipedia,

Exposure therapy involves exposing the target patient to the anxiety source or its context without the intention to cause any danger. Doing so is thought to help them overcome their anxiety or distress.

Psychologists call this "titrated exposure" which is actually the very precise meaning you are looking for, but it's so precise and technical as to not be in common usage.

Titration is the process of gradually and in a measured way increasing something. It particularly describes adding one chemical to another in small, measured increments, in order to determine the precise point at which some state is reached - such as e.g. a neutral ph.

As a technique, this is used to e.g. slowly and incrementally increase somebody's exposure to a traumatic memory or a phobia in order to help them learn acceptance.

I recommend using the word:

adapt VERB If you adapt to a new situation or adapt yourself to it, you change your ideas or behaviour in order to deal with it successfully.

Its derivative noun is "adaptation," thus you can say:

"I spent a couple of nights in a dark room as kind of [adaptation]. I fear the dark, you know"

In Medicine, we call it "exposure therapy", a technique that involves carefully exposing the patient to the obejct causing fear or anxiety.

From Wikipedia Exposure therapy is a technique in behavior therapy to treat anxiety disorders.

• Exposure therapy involves exposing the target patient to the anxiety source or its context without the intention to cause any danger. Doing so is thought to help them overcome their anxiety or distress. Numerous studies have demonstrated its effectiveness in the treatment of disorders such as generalized anxiety disorder, social anxiety disorder, obsessive-compulsive disorder, Post-Traumatic Stress Disorder, and specific phobias.