Notice that from the problem statement it follows that the same number of participants arrived from each company. Indeed, if there were two different numbers of participants from some two companies, they would have a different number of competitors. Let $m$ be the number of companies participating in the conference, each of them having $k$ members. Then, the number of competitors of each participant of the conference equals $(m-1)k$. It is equally easy to devise that the total number of participants in the conference is $mk$. Therefore, it should hold that $(m-1)k = 2018$, or $mk = 2018 + k$. Thus, the largest number of participants will be at the highest value of $k$, and this value is $k = 2018$ when $m = 2$. Thus, the maximum number of participants of the conference is $4036$.