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

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.

Brainfuck, 15 bytes

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

Pretty straightforward implementation, shifting the memory space by 1 for each input char. Requires an interpreter that gives 0 on EOF.

Try it online!

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"]

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.

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.

MATLAB, 23 bytes

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

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

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

Python, 39 bytes

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

Test it on Ideone.

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'

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.

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>

Python, 40 bytes

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

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
}

LINQPad w/ C#, 82 bytes

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

Single output operation.

PHP, 68 bytes

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

PowerShell v2+, 36 bytes

-join([char[]]$args[0]|%{"$_"*++$i})

Takes input $args[0], explicitly casts it as a char array, sends that into a loop |%{...}. Each iteration we take the current letter/character "$_" and use the * overloaded operator to concatenate the string pre-incremented $i times. The result of each loop iteration is encapsulated in parens to form an array and then -joined together to form a string. That string is left on the pipeline and output is implicit.

Examples

PS C:\Tools\Scripts\golfing> .\elasticize-a-word.ps1 Why
Whhyyy

PS C:\Tools\Scripts\golfing> .\elasticize-a-word.ps1 SKype
SKKyyyppppeeeee

PS C:\Tools\Scripts\golfing> .\elasticize-a-word.ps1 LobbY
LoobbbbbbbYYYYY

PS C:\Tools\Scripts\golfing> .\elasticize-a-word.ps1 'a b'
a  bbb

05AB1E, 6 bytes

vN>Fy?

Explained

v       # for each char in string
 N>F    # index+1 number of times do
    y?  # print current char

Try it online

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.

C#, 81 Bytes

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

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

Retina, 62 bytes

It wasn't as easy or short as I thought it'd be. Note that the code contains no spaces. They are all tabs (which are rendered incorrectly here), and the last line is blank.

.*
$0¶ ¶
{+`^(.)(.*)¶    (.*¶.*)
$1$2    ¶$3$1
(   +)¶
¶   $1
}`^.

    |¶

Try it online

NARS2000, 6 bytes

⍳∘⍴/⊙⊢

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

K/Kona, 14 bytes

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

Usage:

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

Julia, 34 bytes

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

Try it online!

Python 2.7 - 47 Bytes

''.join([s[i-1]*i for i in range(1, len(s)+1)])

where 's' is the given string Output:

welcome: weelllccccooooommmmmmeeeeeee
00004:   000000000044444
Why:     Whhyyy
SKype:   SKKyyyppppeeeee
A and B: A  aaannnnddddd      BBBBBBB

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 :-).