Minimal Power Root
Perl 6, 49 bytes
{grep {($_,{$_**.comb.min}...*==*).tail>$_},1..*}
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Returns an infinite sequence. I suppose that the following 45 byte version works, too, but I can't prove that the fixed point is always found after n iterations.
{grep {($_,{$_**.comb.min}...*)[$_]>$_},3..*}
05AB1E, 8 bytes
Generates the nth number 1-indexed
µNÐΔWm}‹
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Explanation
µ # run until counter equals input
NÐ # push 3 copies of the current iteration index (1-based)
Δ } # run this code until the result no longer changes
Wm # raise the number to the power of its minimum digit
‹ # check if greater than the index
Optionally as an infinite list at the same byte count:
∞ʒDΔWm}‹
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J, 41 39 37 bytes
(>:[echo^:(<(^0".@{/:~@":)^:_))^:_]1x
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This one is a full program printing the infinite sequence. A very rare occasion where a full program beats a verb in J.
How it works
(>:[echo^:(<mpi_fix))^:_]1x Using the mpi_fix below; it finds the MPI fixpoint
(<mpi_fix) Is mpi_fix greater than the input?
echo^: If so, apply echo; do nothing otherwise
echo returns an empty array
>:[ Discard the above and return input+1
( )^:_ Repeat the above infinitely (increment has no fixpoint)
]1x starting from arbitrary-precision number 1
J, 41 39 bytes
>:^:(>:(^0".@{/:~@":)^:_)^:_@>:@]^:[&0x
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A monadic verb. Given a 1-based index, returns the number at that index. The footer checks that first 20 terms are correct.
Reading the word "fixpoint", I immediately thought "Oh yeah, ^:_
will do the great job." Then I ended up with this abomination of angry and sad faces. And it's not even a train, it's a single verb.
Ungolfed & How it works
nth_term =: >:^:(>:(^0".@{/:~@":)^:_)^:_@>:@]^:[&0x
mpi =: ^0".@{/:~@": Find the MPI
/:~@": Sort the string representation
0 { Take first item
".@ Convert back to number
^ Raise the input to the power of above
mpi_fix =: mpi^:_ Find the MPI fixpoint
next_term =: >:^:(>:mpi_fix)^:_@>: Given a number, find the next term
@>: Increment once, and then...
>:mpi_fix Is mpi_fix not greater than input?
>:^: ^:_ Increment while the above is true
nth_term =: next_term@]^:[&0x Given one-based index, find the nth term
next_term@] Apply next_term monadically
^:[ n times
&0x to the starting value of zero
The arbitrary-precision integer 0x
is needed to compute the fixpoint accurately, e.g. of the number 6.