How to handle newly discovered problems in second and third round reviews For the first time, I had to review a revision of an academic paper (I have already reviewed several other papers, but they were all rejected after the first round). While the authors had clearly improved the paper (after the first round, major revisions were requested by the editor (and myself)), several major issues mentioned in the first round of reviews were still not (or not correctly) addressed. Furthermore, due to the improvement in writing and structure, the paper was easier to read, and I was able to identify several flaws or strong limitations that I did not report in the first round. Consequently, I, again, recommended major revisions (while I hesitated with rejection due to the major flaws). 

After having submitted my review, I received the reviews of the two others reviewers. To my surprise, both were positive (1 line comment such as "Issues have been addressed and I have no further comment") and the editor requested minor revisions.

In view of this, I am wondering if I have misunderstood how I should review a paper after revision as I did not make a real difference between round 1 and 2. Therefore my question is how far can/should we go in reviewing a paper? If at each round, improvements are made but new flaws or mistakes are spotted, should we stop mentioning them at some point? 
This process might be theoretically infinite. Furthermore, as an author, I know that several rounds of reviews are exhausting and stressful so I do not want to be "that reviewer" (the one who is too picky and who completely slow down the publication process).
(1) The comments of other reviewers should not bother you; often reviewers will be from mutually exclusive fields, so as to provide better representation. You should stand by your professional opinion irrespective of the others.

(2) I firmly believe that you should spend a lot of time on first and second review. If you are finding new and significant errors beyond these rounds, it means you didn't do your job rigorously before.



EDIT: Several comments highlight the valid point that sometimes revisions are required just to make the paper readable. My position on this is that if more than 2 'readability-reviews' are required, the authors are either unable to use the language adequately (and should be directed to a suitable resource- a reviewer is meant to check technical content, not be a style coach), or are unwilling to make these suggestions; in which case there is no point asking once again.



Comments/questions regarding your previous objections are ok, but entirely new objections in third or subsequent rounds leave a bitter taste and lead to a lot if time being wasted.