To exponential digit growth and beyond!
JavaScript (Node.js), 40 bytes

Thanks to @Neil for saving 5 bytes on this version and 2 bytes on the BigInt version

Takes input as (t)(base), where \$t\$ is 1-indexed.



n=&gt;g=(b,x=11)=&gt;--n?g(b,+x.toString(b)):x


Try it online!



JavaScript (Node.js), 48 bytes (BigInt version)

Takes input as (t)(base), where \$t\$ is 1-indexed. Returns a BigInt.



n=&gt;g=(b,x=11n)=&gt;--n?g(b,BigInt(x.toString(b))):x


Try it online!