RNA polymerase that catalyzes the synthesis of short RNA molecules used as primers for DNA polymerase during DNA replication. Also part of the exosome, which is a complex involved in RNA degradation. Acts as a poly(A)-binding protein that enhances the interaction between heteromeric, adenine-rich transcripts and the exosome. ssDNA + n NTP = ssDNA/pppN(pN)n-1 hybrid + (n-1) diphosphate. Binds two Mg(2+) per subunit. Forms a ternary complex with MCM helicase and DNA. Component of the archaeal exosome complex. Belongs to the archaeal DnaG primase family.