A seemingly endless controversy that has already lasted over a decade surrounds Yaroslav Sergeyev's proposed theory of infinity. He has over twenty publications on this topic. Critics maintain that his theory is inconsistent, with whatever valid material actually derived from earlier rigorous sources. He also has some backers including one that posted an answer at MO.
Complete disclosure: I am one of the authors of this 2017 publication in Foundations of Science critical of Sergeyev's production.
Some other possibilities are mentioned in this question.