the ancient Egyptians had no such concept. The king was simply the incarnation of Horus, so there would be no Ancient Egyptian phrase or word to describe it.
There are a number of Egyptian dictionaries. Two of those available online are:
The standard text for Middle Egyptian grammar is still Sir Alan Gardiner's Egyptian Grammar.
I was intrigued by the unusual wording you used in the phrases "Royal road of life" or “Royal trail of life”, and the description you gave in the comments:
The road of a person to kinghood. In the beginning, the person has no knowledge, is a fool, in the end, he possesses all knowledge, became a "king" so to speak. Or another (easier) approach: A person who was a slave, but on his road of life, he learns and collects experiences, triumphs over his weaknesses and becomes a king. But a king of the world, a king who triumphed over himself.
so I did some further research.
The Ancient Egyptian word or phrase for that concept
Firstly, to be completely clear, the ancient Egyptians had no such concept. The king was simply the incarnation of Horus, so there would be no Ancient Egyptian phrase or word to describe it.
There are a number of Egyptian dictionaries. Two of those available online are Faulkner's Middle Egyptian dictionary, and Paul Dickson's Dictionary of Middle Egyptian.
One problem you might find with using these is that the words are arranged in the order of what you might think of as the Ancient Egyptian 'alphabet' - the hieroglyphs that represent a single consonant ('Uniliteral hieroglyphs'). To make effective use of the dictionaries, you have to learn the order of those consonants.
If you are interested, the standard text for Middle Egyptian grammar is still Sir Alan Gardiner's Egyptian Grammar.
The origin of the phrase
So I wondered where that phrase might have originated. After all, as far as I'm aware they aren't commonly used phrases (at least in English).
A Google search revealed that it is commonly used on sites relating to tarot.
A little more research suggests that the earliest use may have been by Antoine Court (who styled himself Antoine Court de Gébelin), in his 1781 book Monde primitif analysé et comparé avec le monde moderne considéré dans divers objects concernant l'histoire, le blason, les monnoies, les jeux, les Voyages des Phéniciens autour du Monde, les langues Américaines, etc. ou dissertations mêlées where he wrote:
Le nom de ce Jeu est pur Egyptien : il est compose du mot TAR , qui signifie voie, chemin, & du mot RO , ROS, ROG, qui signifie Roi, Royal. C’est , mot-à-mot, le chemin Royal de la vie.
or (my translation)
The name of this game is pure Egyptian: it is composed of the word TAR, which means 'path', or 'trail' and the word RO, ROS, ROG, which means 'King' or 'Royal'. It is, word by word, the Royal trail of life.
(I would normally have translated the highlighted text as 'the Royal way of life', but I have rendered it here as I have found it rendered on those tarot sites I described above).
This is certainly the text most commonly cited by those sites.
Credibility of Antoine Court de Gébelin's claim
So how credible are his claims?
Frankly, they're nonsense.
It is worth noting that Court's book was published in 1781. That is decades before the work of Thomas Young and Jean-François Champollion began the process that led to the understanding of Egyptian hieroglyphs and the language of the ancient Egyptians.
Today we can read the ancient Egyptian hieroglyphs.
The ancient Egyptian word for 'king' was nsw or nsw bity (king of Upper and Lower Egypt), - nothing like 'RO' as claimed by Court.
The word for 'road' could be written in a number of ways. For example, as wAt or mTn . However, none of them can transliterate as anything like 'TAR', as Court claimed.
So Antoine Court de Gébelin's claims were wrong, but he was on pretty safe ground since nobody else could read the Egyptian hieroglyphs at that time either.
The minor detail that we can now read Egyptian hieroglyphs, and we can therefore say categorically that Court was wrong, doesn't seem to stopped the claim from being repeated.