Output integers in negative order, increase the maximum integer everytime

05AB1E, 6 bytes

L€LíJJ

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Explanation

Example input 4

L       # range [1 ... input]
        # STACK: [1,2,3,4]
 €L     # map: range
        # STACK: [[1],[1,2],[1,2,3],[1,2,3,4]]
   í    # reverse each
        # STACK: [[1],[2,1],[3,2,1],[4,3,2,1]]
    J   # join inner lists
        # STACK: ['1','21','321','4321']
     J  # join list
        # OUTPUT: 1213214321

JavaScript (ES6), 37 bytes

f=(n,k=1)=>k>n?n--?f(n):'':f(n,k+1)+k

Demo

f=(n,k=1)=>k>n?n--?f(n):'':f(n,k+1)+k

console.log(f(6))

Alternate method for n < 10, 34 bytes (non-competing)

f=(n,s='1')=>--n?s+f(n,++s[0]+s):s

In JavaScript, strings are immutable. Therefore, it's impossible to alter the content of the Nth character of a string s by assigning a new value to s[N].

However, the expression ++s[N] is valid and does evaluate as one would expect, even if the string remains unchanged. For instance:

++"1"[0] // equals 2

And by extension:

s = "21"
++s[0] + s // equals "321"

Jelly, 5 bytes

RRUVV

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Formula not mine.

I suspect there is too much going on here...

[ANSWER ACCEPTED] I'd have given some 5 rep to Dennis, but this is not Reputation Exchange. Dennis showed me the VV behavior. To my surprise, this is shorter than 05AB1E.