Catalyzes the transfer of one adenosine molecule from an ATP to an mRNA poly(A) tail bearing a 3'-OH terminal group in an ATP hydrolysis-dependent manner. May be involved in maintaining the translation efficiency of at least some genes through preventing degradation of their mRNAs. Prefers RNA molecules that are adenosine-rich close to 3'-end. In addition, may inhibit cell proliferation and cell cycle progression through ubiquitination of beta-catenin/CTNNB1. ATP + RNA(n) = diphosphate + RNA(n)-3'-adenine ribonucleotide Belongs to the TENT family. It is uncertain whether Met-1 or Met-2 is the initiator.