Mathematica, 113 90 bytes

Thanks to Martin Ender for two suggestions that reduced the length by over 20%!

Showing off the high-level commands in Mathematica.

Nest[Flatten@(Characters/@{RomanNumeral@#,#2}&@@@(Reverse@#&@@@Tally/@Split@#))&,{"I"},#]& 

A pure function, taking an argument N and outputting the Nth element of this (0-indexed) sequence, as a list of characters. Spread out a bit:

Nest[ Flatten@ (Characters/@ {RomanNumeral@#,#2}& @@@   (Reverse@# & @@@ Tally /@ Split@#)  ) & , {"I"}, #] & 

The outer Nest iterates the middle four-line function, starting on {"I"}, N times. Line 4 splits the character list of the input Roman numeral into runs of like characters, counts each run with Tally, and puts the counts before the characters they're counting. Line 3 renders the counts as Roman numerals, then splits those Roman numerals up into lists of characters. The Flatten command reduces the whole list-of-lists to a one-dimensional list.

Here's the initial version:

Nest[ "" <> Flatten[{RomanNumeral@#[[1]], #[[2]]} & /@ (Reverse@#[[1]] & /@ Tally /@ Split@Characters@#)] &, "I", #] & 

Mathematica, 113 90 83 bytes

Thanks to Martin Ender for suggestions that reduced the length by over 25%!

Showing off the high-level commands in Mathematica.

Nest[Flatten[Characters@{RomanNumeral@#,#2}&@@@Reverse@@@Tally/@Split@#]&,{"I"},#]& 

A pure function, taking an argument N and outputting the Nth element of this (0-indexed) sequence, as a list of characters. Spread out a bit:

Nest[ Flatten[ Characters @ {RomanNumeral@#,#2}& @@@ Reverse @@@ Tally /@ Split@ #  ]& , {"I"}, #]& 

The outer Nest iterates the middle four-line function, starting on {"I"}, N times. Line 4 splits the character list of the input Roman numeral into runs of like characters, counts each run with Tally, and puts the counts before the characters they're counting. Line 3 renders the counts as Roman numerals, then splits those Roman numerals up into lists of characters. The Flatten command reduces the whole list-of-lists to a one-dimensional list.

Here's the initial version:

Nest[ "" <> Flatten[{RomanNumeral@#[[1]], #[[2]]} & /@ (Reverse@#[[1]] & /@ Tally /@ Split@Characters@#)] &, "I", #] & 

Mathematica, 113 bytes

Didn't golf much—just showing off the high-level commands in Mathematica.

Nest[""<>Flatten[({RomanNumeral[#1[[1]]],#1[[2]]}&)/@(Reverse[#1[[1]]]&)/@Tally/@Split[Characters[#1]]]&,"I",#1]& 

A pure function, taking an argument N and outputting the Nth element of this (0-indexed) sequence. Spread out a bit:

Nest[ "" <> Flatten[{RomanNumeral@#[[1]], #[[2]]} & /@  (Reverse@#[[1]] & /@    Tally /@  Split@Characters@#)] &, "I", #] & 

The outer Nest iterates the middle four-line function, starting on "I", N times. Line 5 splits the characters of the input Roman numeral into runs of like characters; Line 4 counts each run; Line 3 puts the counts before the letters they're counting. Line 2 renders the counts as Roman numerals, and then all the Roman numeral strings get concatenated.

Mathematica, 113 90 bytes

Thanks to Martin Ender for two suggestions that reduced the length by over 20%!

Showing off the high-level commands in Mathematica.

Nest[Flatten@(Characters/@{RomanNumeral@#,#2}&@@@(Reverse@#&@@@Tally/@Split@#))&,{"I"},#]& 

A pure function, taking an argument N and outputting the Nth element of this (0-indexed) sequence, as a list of characters. Spread out a bit:

Nest[ Flatten@  (Characters/@ {RomanNumeral@#,#2}& @@@  (Reverse@# & @@@ Tally /@ Split@#)  ) & , {"I"}, #] & 

The outer Nest iterates the middle four-line function, starting on {"I"}, N times. Line 4 splits the character list of the input Roman numeral into runs of like characters, counts each run with Tally, and puts the counts before the characters they're counting. Line 3 renders the counts as Roman numerals, then splits those Roman numerals up into lists of characters. The Flatten command reduces the whole list-of-lists to a one-dimensional list.

Here's the initial version:

Nest[ "" <> Flatten[{RomanNumeral@#[[1]], #[[2]]} & /@ (Reverse@#[[1]] & /@ Tally /@ Split@Characters@#)] &, "I", #] &