A Rajarshi is a king who turned into a royal sage. A Rajarshi will not have to leave the kingship to become rishi as in the example of Vishwamitra (who later advanced to becoming a Brahmarishi) but still while ruling the kingdom has reached a state of rishi and has attained self-realization. They still perform the kshatriya responsibility, and remain similar to most rishis, maharishis and brahmarishis descendants in their level of spiritual knowledge.
Namely there are four types of Rishis mentioned in Hinduism and Vedas.
In Vedas and Holy scriptures of Hindusim, the name and Rajarshi has its own meaning and it's widely being applied for few ancient historical Hindu Legends who always and still now stood as examples of sacrifices for achieving a greater cause for the humanity and the prevalent societies. There are the mention of namely two Rajarshi in Vedic History.
Rajarshi Mudgal wrote 1 Upnishad out of 108 Upnishads named Mudgalupnishad. Mudgalupnishad is unique amongst upanishads, as it mainly contains information and praise concerning Lord Ganesha, and how to pray and perform Yajna and Puja of Lord Ganpati.
The great sage strongly believed in simple living high thinking and had a high level patience amongst other Rishis. That's why still today the persons belonged to Mudgal Gotra are considered highly ethical and had infinite patience in compare to average.
Rajarshi Mudgal was also been cited for the infamous curse on his wife. He cursed his wife that in next birth she will have Five Husbands. And due to that effect of his spell, his wife was born as Draupadi, the daughter of King Drupada in her next birth.
The great sage Vishvamitra was also called Rajarshi who later went on to become Brahmarshi.
Another famous figure named Rajarshi in somehow present era was Purusottam Das Tandon. He was the one who is regarded as the person to make Hindi the Official Language of Independent India. He was awarded Bharat Ratna (India's Highest Civilian Award) in year 1961.
Also Chhatrapati Shahu Maharaj of Kolhapur(26 June 1874 – 6 May 1922) is known as Rajarshi Shahu. During his rule in Kolhapur State from 1894 till death in 1922, he worked for lower caste, education to all, social reforms, reservation, agriculture reforms & much more.