J, 4 bytes

#~#\

Usage

   f =: #~#\
   f 'Why'
Whhyyy
   f 'SKype'
SKKyyyppppeeeee
   f 'LobbY'
LoobbbbbbbYYYYY
   f 'A and B'
A  aaannnnddddd      BBBBBBB

Explanation

#~#\  Input: s
  #\  Computes the length of each prefix of s
      This forms the range [1, 2, ..., len(s)]
#~    For each value in the range, copy the character at the
      corresponding index that many times
      Return the created string

Brainfuck, 15 bytes

,[>+[->+<<.>],]

Pretty straightforward implementation, shifting the memory space by 1 for each input char. Requires an interpreter that gives 0 on EOF, and 32-bit/arbitrary precision cells for inputs longer than 255 chars.

Try it online! (Note: TIO uses 8-bit cells)

Haskell, 29 bytes

concat.zipWith replicate[1..]

Usage example: concat.zipWith replicate[1..] $ "SKype" -> "SKKyyyppppeeeee".

replicate n c makes n copies of c and concat makes a single list out of all the sublists.

Java, 158 121 bytes

Saved a whopping 37 bytes thanks to Kevin Cruijssen!

interface a{static void main(String[]A){int b=0,B;for(char c:A[0].toCharArray())for(B=b+++2;--B>0;)System.out.print(c);}}

As a bonus, this program can handle all Unicode characters in the existence, including the control characters located at the very end of Basic Multilingual Plane.

Perl, 16 bytes

s/./$&x$+[0]/ge

+1 byte for the -p flag.

s/./        /    find every character
             g   globally
              e  and replace with the eval'd result of
    $&           the matched string
      x          repeated
       $+[0]     by the index of the character after the match

CJam, 9 8 7 bytes

Thanks to jimmy23013 for saving 1 byte.

Sl+eee~

Test it here.

Explanation

Using the LobbY example:

                                      Stack:
S    e# Push space.                   [" "]
l    e# Read input.                   [" " "LobbY"]
+    e# Append.                       [" LobbY"]
ee   e# Enumerate.                    [[[0 ' ] [1 'L] [2 'o] [3 'b] [4 'b] [5 'Y]]]
e~   e# Run-length decode.            ["LoobbbbbbbYYYYY"]

Python, 39 bytes

f=lambda s:s and f(s[:-1])+s[-1]*len(s)

Test it on Ideone.

APL (8)

{⍵/⍨⍳⍴⍵}

I.e.:

      {⍵/⍨⍳⍴⍵} ¨  'Why' 'SKype' 'LobbY'
┌──────┬───────────────┬───────────────┐
│Whhyyy│SKKyyyppppeeeee│LoobbbbbbbYYYYY│
└──────┴───────────────┴───────────────┘

Explanation:

• ⍴⍵: length of given vector
• ⍳: numbers 1..N
• ⍵/⍨: replicate each element in ⍵ N times.

MATLAB, 45 bytes

g=@(m)sort(m(m>0));@(s)s(g(hankel(1:nnz(s))))

Explanation: The key is hankel, which produces a Hankel matrix of a given vector. From this matrix, we can extract a vector of indices, which defines which character of the string is at which position in the output. E.g. hankel(1:4) produces following matrix:

 1  2  3  4
 2  3  4  0
 3  4  0  0
 4  0  0  0

From this matrix we can extrac the vector 1,2,2,3,3,3,4,4,4,4,4. This vector allows us to output the first character of the string once, the second one twice e.t.c.

Javascript ES6, 39 bytes

x=>x.replace(/./g,(y,i)=>y+y.repeat(i))

Same length, but more fun:

x=>x.replace(i=/./g,y=>y.repeat(i=-~i))

Snippet demo:

f= x=>x.replace(/./g,(y,i)=>y+y.repeat(i))
run.onclick=_=>output.textContent=f(input.value)

<input id="input" value="SKype">
<button id="run">Go</button>
<pre id="output"></pre>

NARS2000, 6 chars = 12 bytes

⍳∘⍴/⊙⊢

⍳∘⍴ enumeration of the argument... (indices of its length)
/⊙ replicates the elements of...
⊢ the unmodified argument

MATLAB, 23 bytes

@(x)repelem(x,1:nnz(x))

Creates an anonymous function ans that can be called using ans('stringtoelacticize')

K/Kona, 14 bytes

{,/(1+!#x)#'x}

Usage:

k){,/(1+!#x)#'x}"A and B"
"A  aaannnnddddd      BBBBBBB"

Perl 6,  22 20  19 bytes

{S:g/(.)/{$0 x$/.to}/}
{S:g[(.)]=$0 x$/.to}
{[~] .comb Zx 1..*}

Explanation:

{          # implicit parameter $_
  [~]      # string concatenate the following list
    .comb  # the NFG characters from $_
    Z[x]   # zip combined using the string repetition operator
    1 .. * # 1 to infinity
}

PHP, 68 bytes

<?php foreach(str_split($argv[1])as$i=>$a)echo str_repeat($a,$i+1);

Actually, 7 bytes

' +ñ♂πΣ

Try it online!

Explanation:

' +ñ♂πΣ
' +      prepend a space
   ñ     enumerate ("abc" -> [[0, 'a'], [1, 'b'], [2, 'c']])
    ♂π   map: for each character, repeat it n times
      Σ  concatenate

Pyth - 5 bytes

1 byte saved thanks to @FryAmTheEggman.

s*VSl

Test Suite.

Python 3, 48 47 bytes

Thanks to mego for saving a byte with the -~i trick.

lambda s:''.join(c*-~i for i,c in enumerate(s))

This is mostly self-explanatory. One thing for those not versed in Python: The * operator is overloaded to act like Perl's x operator, repeating its string argument the number of times specified by its numeric argument. E.g. 'foo' * 3 == 'foofoofoo'

C#, 81 Bytes

void f(string s){for(int i=0;i<s.Length;i++)Console.Write(new String(s[i],i+1));}

Brachylog, 13 bytes

:ImC,0:Ie,Cw\

This prints the result to STDOUT.

Explanation

This is a good example of exploiting backtracking to loop.

:ImC            C is the Ith character of the Input
    ,
     0:Ie       Unify an implicit variable with an integer between 0 and I
         ,
          Cw    Write C to STDOUT
            \   False, trigger backtracking. It will go back to 0:Ie and unify the implicit
                variable with another integer, until all integers were used. After that, it
                will backtrack to :ImC and unify I and C with the next character.

MATL, 5 bytes

tn:Y"

Try it Online

Explanation

    % Implictly grab input as a string
tn  % Duplicate and compute the length (N)
:   % Create an array from [1...N]
Y"  % Perform run-length decoding to elacticize the string
    % Implicitly display the result

Python, 40 bytes

f=lambda s,i=1:s and s[0]*i+f(s[1:],i+1)

Julia, 34 bytes

!s=s>""?!s[1:(e=end)-1]*s[e:e]^e:s

Try it online!

VBA, 75 bytes

Function e(s):For a=1 To Len(s):e=e &String(a,Mid(s,a,1)):Next:End Function

Call as e.g. a user function in a spreadsheet.

=e(A1)

┌─────────┬───────────────┐
│   SKype │SKKyyyppppeeeee│
└─────────┴───────────────┘

It truncates if you feed it its own output a few times :-).

TSQL, 97 bytes

Golfed:

DECLARE @x varchar(max)='Lobby'
DECLARE @ int=LEN(@x)WHILE @>0SELECT
@x=STUFF(@x,@,1,REPLICATE(SUBSTRING(@x,@,1),@)),@-=1PRINT @x

Ungolfed:

DECLARE @x varchar(max)='Lobby'

DECLARE @ int=LEN(@x)
WHILE @>0
  SELECT 
    @x=STUFF(@x,@,1,REPLICATE(SUBSTRING(@x,@,1),@)),
    @-=1

PRINT @x

Try it online

Julia, 32 bytes

!s=join(split(s[k=1:end],"").^k)

Unlike Dennis's solution, this is not recursive. split with argument "" separates the string into an array of strings of length 1. The [k=1:end] is a trick to create a range from 1 to the number of characters in the string, and this range is used to concatenate n copies of the n-th character. join then recombines the array of strings into a single string, in order.

Usage example: !"SKype"

Mathematica, 46 bytes

""<>Table@@@(#^Range@Length@#&@Characters[#])&

Unnamed function. Takes the characters of the input string and raises them to the power of their position in the string (i.e. "abc" becomes {"a"^1, "b"^2, "c"^3}). Yes, MMA don't give a shit 'bout types :-)

The FullForm of the above list elements is Power["a", 1], etc.
Table@@@ acts on the list, replacing the head of each element (in this case Power) with Table.
Results in {Table["a"], Table["b", 2], Table["c", 3]} (because "a"^1 -> "a").
This evaluates to {"a", {"b", "b"}, {"c", "c", "c"}} Finally the infix concatenation operator <> concatenates this with the empty string.

I exploited the fact that Power is Listable, i.e. it automatically threads over corresponding elements of lists, but not orderless (because "a" + 1 would evaluate to Plus[1, "a"]).

LINQPad w/ C#, 82 bytes

void m(string s){Console.Write(s.SelectMany((x,i)=>new string(x,++i)).ToArray());}

Single output operation.

C, 84 bytes

main(c,v,i,j)char**v;{for(i=0;i<strlen(v[1]);++i)for(j=0;j<=i;++j)putchar(v[1][i]);}

This should compile on gcc with no flags. Input is taken through the first command-line argument. E.g.

$ ./elasticize Why
Whhyyy
$ ./elasticize SKype
SKKyyyppppeeeee
$ ./elasticize LobbY
LoobbbbbbbYYYYY
$ ./elasticize A and B
A  aaannnnddddd      BBBBBBB

Ungolfed:

int main(int argc, char** argv) {
    int i, j;
    for(i = 0; i < strlen(argv[1]); ++i) {
        for(j = 0; j <= i; ++j) {
            putchar(v[1][i]);
        }
    }
}