The Earth is a rocky planet formed by the accretion/agglomeration of cosmic dust and ice. One might argue that the first rock of Earth was an agglomeration, and once a critical mass had been reached, the agglomerated core heated and began to melt creating igneous rock, assisted by the heat of bombardment from more cosmic rocks and of radioactive decay.
The idea of the "rock cycle" has prerequisites to even be meaningful. These include:
Without all of these, the concept of the rock cycle (as introduced in textbooks) really is not applicable. (You might suppose there were other 'rock cycles' in the very distant past but that is not what is discussed in undergraduate textbooks.)
The Earth is older than (1) (2) or (3) - so when these preconditions were not in place there was no rock cycle. The three preconditions also did not "turn on" at exactly the same time. It took a long time for the Earth's crust to form (When and how did continental crust form?). Oceans and lakes appeared later (When did oceans form on Earth?). Plate tectonics may have began early (When did plate tectonics begin?).
The point is that the planetary conditions necessary for the rock cycle we see evidence for in the geology of the last billion years or so evolved over much time. Earth was very different in the distant past.
This is the reason why there are no scientists working on the question: "What was the first rock in the rock cycle?" Scientists would like to answer big questions, such as:
Rocks are assumed to be solids and not a fluid part of a melt.
Bowens Reaction Series implies that the first rocks on earth would have been Olivine Polymorphs.
MgSiO3-perovskite and MgO (periclase) are likely candidates from observations at the Earths Moho.
As far as the "rock cycle" goes; these first rocks would be classified as igneous.