Perl 6, 31 bytes

{$^a;first {$a**$_~~/$a/},2..*}

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R, 69 bytes

function(n,i=2){while(!grepl(?n, ?n^i))i=i+1;cat(i)}

`?`=as.character

Anonymous function.

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JavaScript (ES6),  41  40 bytes

Saved 1 byte thanks to @Shaggy

Takes input as a BigInt literal.

n=>(g=x=>`${x*=n}`.match(n)?2:-~g(x))(n)

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Python 2, 42 41 bytes

^{-1 byte thanks to Ørjan Johansen (returning y directely)}

f=lambda x,y=2:y*(`x`in`x**y`)or f(x,y+1)

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Explanation/Ungolfed

Recursive function trying from $2,3dots$ until we succeed:

# Start recursion with y=2

def f(x,y=2):

    # If we succeed, we arrived at the desired y

    if `x` in `x**y`:

        return y

    # Else we try with next y

    else:

        return f(x, y+1)

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APL (Dyalog Unicode), 25 23 17 bytes

-2 bytes thanks to @Erik the Outgolfer

-6 bytes thanks to @ngn

⊢⍟×⍣(∨/(⍕÷)⍷∘⍕⊣)⍨

My mind is blown.

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APL (Dyalog Unicode), 25 bytes

2∘{∨/(⍕⍵)⍷⍕⍵*⍺:⍺⋄(⍺+1)∇⍵}

This function is composed with the left argument 2.

2∘{∨/(⍕⍵)⍷⍕⍵*⍺:⍺⋄(⍺+1)∇⍵}

   ∨/                     At any point

     (⍕⍵)                 can the string representation of ⍵

         ⍷                be found in

          ⍕⍵*⍺            the string representation of ⍵ to the power of ⍺? (starting at 2)

              :⍺          If so, return ⍺

                 (⍺+1)∇⍵  Else, return the result of the function with ⍺+1

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Pyth, 9 bytes

f}`Q`^QT2

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Jelly, 7 bytes

2ẇ*¥@1#

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05AB1E, 7 bytes

∞>.Δm¹å

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Ruby, 37 bytes

->n,i=2{i+=1until/#{n}/=~"#{n**i}";i}

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Japt, 10 bytes

@pX søU}a2

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JavaScript (Node.js), 45 bytes

Test cases taken from @Arnauld's answer

a=>eval("for(i=1n;!(''+a**++i).match(a););i")

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Charcoal, 19 bytes

Ｗ∨‹Ｌυ²¬№ＩΠυθ⊞υＩθＩＬυ

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

Ｗ∨‹Ｌυ²¬№ＩΠυθ⊞

Repeat until the the list length is at least 2 and its product contains the input...

⊞υＩθ

... cast the input to integer and push it to the list.

ＩＬυ

Cast the length of the list to string and implicitly print it.

Python 3, 63 58 bytes

def f(n,e=2):

	while str(n)not in str(n**e):e+=1

	return e

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Python2 would probably be shorter, but I like using 3. Coming up wiht a lambda is difficult, but I'm trying a few things.

C# (.NET Core), 104 bytes

a=>{for(int i=2;;i++)if(System.Numerics.BigInteger.Pow(a,i).ToString().Contains(a.ToString()))return i;}

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Need to use C#'s BigInteger library, as the standard numeric C# types (int, double, long, ulong, etc.) fail for some larger numbers (including 12, 15, and 17).

Ungolfed:

a => {

    for(int i = 2; ; i++)                           // starting from 2

        if( System.Numerics.BigInteger.Pow(a,i)         // n = a^i

                .ToString().Contains(a.ToString()) )    // if n contains a

            return i;                                       // return i

}

Clean, 99 bytes

import StdEnv,Text,Data.Integer

$n=hd[p\p<-[fromInt 2..]|indexOf(""<+n)(""<+prod(repeatn p n))>=0]

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If it doesn't need to work for giant huge numbers, then

Clean, 64 bytes

import StdEnv,Text

$n=hd[p\p<-[2..]|indexOf(""<+n)(""<+n^p)>=0]

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