There is no definitive answer to this question as it depends on various factors such as how you measure growth (e.g. number of converts, percentage of population, etc.) and what time period you are looking at. However, based on available data, it appears that Islam is the fastest growing religion in Russia.
According to a study by the Pew Research Center, the Muslim share of the population in Russia increased from 10% in 2010 to 14% in 2015. This was due largely to higher birth rates among Muslims than other religious groups, as well as some conversion to Islam. The same study found that Orthodox Christianity was still the largest religion in Russia, with 71% of the population identifying as Orthodox Christian in 2015. However, the Orthodox Christian share of the population had decreased slightly from 77% in 2010.
Other studies have found similar results regarding Islam's growth relative to other religions in Russia. For example, a 2017 study by researchers at Kaunas University of Technology found that between 1989 and 2014, Islam was the only religion to experience significant growth in Russia (measured by both number of adherents and percentage of total population). The same study found that while Orthodox Christianity remained dominant during this time period (with around two-thirds of Russians identifying as Orthodox Christians), there was a decline both in terms of absolute numbers and percentages when compared with earlier periods such as Soviet times.
So overall, it seems clear that Islam is currently experiencing the most rapid growth among all religious groups within Russia's borders.