Husk, 6 bytes

U¡(ṁΣd

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Explanation

U¡(ṁΣd
 ¡(       Iterate the following function on the input:
     d       Split the number into digits
   ṁΣ        Map each digit to its triangular number, then sum the results
U         Take the results of iteration until before the first repeated one

J, ₂₀ 19 bytes

(1#.2!1+,.&.":)^:a:

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Outputs the original number, too.

Explanation

(1#.2!1+,.&.":)^:a:
               ^:a:  Apply until input converges, storing all results in an array
(1#.2!1+,.&.":)      Digitangular sum
        ,.&.":         Split to digits
          &.":           Convert to string, apply left function, then undo conversion
        ,.               Ravel items (make array of singleton arrays of items)
                         This ensures that when cast back to integers, the digits are split.
      1+               Add 1 to each
    2!                 Compute (n choose 2) on each (nth triangular number)
 1#.                   Sum (debase 1)

05AB1E, 6 5 bytes

Δ=SLO

Try it online! Edit: Saved 1 byte thanks to @Emigna. Explanation:

Δ       Repeat until value doesn't change
 =      Print current value
  S     Split into characters
   L    Turn each character into range from 1 to N
    O   Sum

APL (Dyalog), 23 20 17 bytes

3 bytes saved thanks to @ngn

{⍵∪⍨+/∊⍳¨⍎¨⍕⊃⍵}⍣≡

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How?

⍵∪⍨ - prepend current array to the

+/ - sum of

∊ - flattened

⍳¨ - ranges of each

⍎¨⍕ - digit of the

⊃⍵ - previous value

⍣≡ until convergence. The usage of ∪ (union) makes sure after the first 1 is joined, the next will be excluded due to set uniqueness, and the array will converge.

Haskell, 51 47 46 bytes

f 1=[1]
f x=x:f(sum$do d<-show x;[1..read[d]])

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f 1=[1]                     -- if x == 1, return the singleton list [1]
f x=                        -- else
         do d<-show x       --  for each char 'd' in the string representation of x
                            --    (using the list monad)
           [1..read[d]]     --  make a list of [1..d]
                            --    (the list monad concatenates all those lists into a single list)
        sum                 --  sum those numbers
      f                     --  call f on it
    x:                      --  and put x in front of it 

Edit: @H.PWiz saved a byte. Thanks!

Python 2, 62 bytes

f=lambda x:x<2and[1]or[x]+f(sum(-~int(i)*int(i)/2for i in`x`))

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Wolfram Language (Mathematica), 43 bytes

Echo@#>1&&#0@Tr[#(#+1)/2&@IntegerDigits@#]&

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How it works

The expression Tr[#(#+1)/2&@IntegerDigits@#] gives the digitangular counterpart of #. We Echo the input, use short-circuit evaluation with && to stop if we've reached 1, and otherwise recurse on the digitangular counterpart.

Ruby, 60 47 42 bytes

-13 bytes by @JustinMariner

-5 bytes by @GB

->x{p x=x.digits.sum{|k|k*-~k/2}while x>1}

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Wolfram Language (Mathematica), 49 42 bytes

#//.x_:>(#.(#+1)/2&)@IntegerDigits@Echo@x&

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Prints each value on its own line, preceded by >>, and includes the starting number.

R, 70 bytes

f=function(n)"if"(n-1,c(n,f((d=n%/%10^(nchar(n):0)%%10)%*%(d+1)/2)),n)

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Returns the original value as well.

R, 80 bytes

function(n){o={}
while(n-1){n=(d=n%/%10^(nchar(n):0)%%10)%*%(d+1)/2
o=c(o,n)}
o}

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Does not return the original value.

Lua, 91 bytes

function f(n)x=0 for d in((0|n)..""):gmatch"."do x=x+d*(d+1)/2 end print(x)c=x==1or f(x)end

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05AB1E, 20 12 bytes

Saved 2 bytes thanks to caird coinheringaahing

ΔD,þ€iLO}O}}

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Explanation

(old version)

Δþ€iD>*;}OD1›iD,}}1,  Main program
Δ                }    Repeat until there is no changes
 þ                    Push digits of the input number
  €i    }             Apply for each digit
    D>*;              Calculate the triangular number for given digit
         O            Sum all numbers
          D1›iD,}     Print if greater than 1
                  1,  Print 1 at the end

Japt, 19 bytes

A quick bit of golf before bed. Can probably be improved on; will come back to it with fresh eyes in the morning and also add an explanation.

Takes input as a single element array.

Ì¥1?U:ßUpUÌì ®ò xÃx

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JavaScript, 61 57 bytes

f=a=>a-1?([...a+[]].map(b=>a+=b++*b/2,a=0),a)+' '+f(a):''

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Python 2, 69 bytes

def f(x):
 while x>1:
  x=sum(-~int(z)*int(z)/2for z in`x`)
  print x

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Charcoal, 18 bytes

Ｗ⊖Ｉθ«≔ＩΣ⭆θΣ…·0κθθ⸿

Try it online! Link is to verbose version of code. Explanation:

   θ                Input
  Ｉ                 Cast to integer
 ⊖                  Decrement
Ｗ   «               Loop while not zero
         θ          Current value
        ⭆           Map over characters and concatenate
             0      Literal character 0
              κ     Current character
           …·       Inclusive range
          Σ         Concatenate
       Σ            Sum of digits
      Ｉ             Cast to string
     ≔         θ    Assign back to value
                θ   Output latest value
                 ⸿  Move to start of next line

dc, 62 bytes

?sj[0soljZ1-[dljr10r^/10%d1+*2/lo+sod1-r0<i]dsixlopdsj1<b]dsbx

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k, 19 bytes

{+/(+/1+!"I"$)'$x}\

Unsurprisingly works similarly to the already posted APL and J solutions

               $x    cast x (implicit var of lambda) to string
   (         )'      apply composition (function train) to each character of string
    +/1+!"I"$        cast character to number, create list from 1 to n, sum it
 +/                  sum triangular numbers
{                }\  iterate lambda until result converges, appending each result

Jelly, 7 bytes

DRFSµÐĿ

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• DRFSµÐĿ: Full program / monadic link.

• ÐĿ: Loop until results are no longer unique (if something other than 1 would occur twice, then the given input does not have a defined result, since it would never reach 1).

• D: Convert from integer to decimal.

• R: Range (1-indexed). Vectorizes.

• F: Flatten and S: Sum (µ just creates a new monadic chain)

Ohm v2,  9  7 bytes

·Ω}#ΣΣu

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Explanation

          Implicit input as a string
·Ω        Evaluate until the result has already been seen, pushing intermediate results
  }       Split digits
   #      Range from 0 to N
    ΣΣ    Sum
      u   Convert to string

Retina, 21 bytes

;{:G`
.
$*1¶
1
$%`1
1

Try it online! (The outputs of the individual cases aren't well separated, but each output ends with a 1.)

Prints each number on its own line, in order, including the starting number.

Explanation

;{:G`

This is just some configuration of the program. { makes the program loop until it fails to change the result (which happens once we get to 1), : prints number before each iteration, and ; prevents the final result from being printed twice at the end of the program. The G is just my usual way of creating a no-op stage.

.
$*1¶

Convert each digit to unary and put it on its own line.

1
$%`1

Compute the triangular number on each line, by replacing each 1 with its prefix. We could also use M!&`1+ here, which gives us all suffixes of each line.

1

Count all 1s, which sums up all the triangular numbers and converts the result back to decimal.

dc, 31 bytes

[[I~d1+*2/rd0<m+]dsmxpd1<f]dsfx

The function m computes the digitangular function of its input; f repeats this until the result reaches 1.

Note that we use the input radix to extract digits - this means it will work in any base system, not only decimal.

Demo

for i in 23 72 55 78 613 8392 11111 8592025 999999999
    do echo $i '=>' $(dc -e $i'[[I~d1+*2/rd0<m+]dsmxpd1<f]dsfx')
done

23 => 9 45 25 18 37 34 16 22 6 21 4 10 1
72 => 31 7 28 39 51 16 22 6 21 4 10 1
55 => 30 6 21 4 10 1
78 => 64 31 7 28 39 51 16 22 6 21 4 10 1
613 => 28 39 51 16 22 6 21 4 10 1
8392 => 90 45 25 18 37 34 16 22 6 21 4 10 1
11111 => 5 15 16 22 6 21 4 10 1
8592025 => 117 30 6 21 4 10 1
999999999 => 405 25 18 37 34 16 22 6 21 4 10 1

Python 2, 76 bytes

f=lambda k,s=0:k+s<2and[1]or[k]*(s<1)+f(*[k/10,s,s+k%10*(k%10+1)/2][k<1::2])

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Befunge-93, 57 bytes

p&>:25*%:1+*2/00g+00p25*/:!v
  ^         p000_@#-1.::g00_

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