$\frac{\pi}{3}e$ is approximately equal to:
2.84657807422452235515451695651552483167829617858837165395986704339307620371911026919085462323682464797125831417055915588210706253023200799687278781793023471514007199829654447617582233348895922031024237096797835826800351909013449542434666218846881388726409761904569405458137658722823724181557439473446917219291006835569333157358766322654791237966966946623595073544349432678461546724384333992745425590086011913751990685057546632872448319697537470763226921527595140181054405265139836753917775379775697089746424785856104245156874061419780994637730604531480206814256732829792410814871969646790020050463395548624913283274551800217089819065169939172989203287463740049502718515202660522891769528913026965548612122163723732564317175312804182112811958680271600871936194429224533157202322236063849980512815661793560779711542728888881905074320...
While $e$ contains a lot of repeating digits - I'd normally expect this to disappear once $\pi$ is introduced.
It almost looks like the further along you go, the greater the degree of repetition.
Is this just due to an interesting interaction that occurs only with decimal approximations of pi and e, or is there more going on?