Teaser 2916: Pointless batting averages

From The Sunday Times, 12th August 2018 [link]

When my son was chosen to play cricket for his School First XI, I kept a record of the number of runs he scored in each of his first five innings. After each innings I calculated his batting average (the total number of runs scored so far divided by the number of innings) and found an interesting pattern:

(i) Each score was under 30
(ii) They [the scores] were all different
(iii) Each of the five averages was a whole number

When he asked me how he could maintain this pattern with his sixth innings, I was able to tell him the smallest score that would achieve this.

What is the largest number this could have been?

[teaser2916]