How can I easily double any size number in my head?

I usually look for an easy calculation that is close by the original one.

For example:

$2 \times 183 = 2 \times 180 + 6=366$

or

$ 2\times 1481= 2 \times 1500 - 38 = 2960+2$

It may help to partition the number into smaller numbers that lead with a number less than 5. The idea being that you can double each of these smaller numbers independently then combine their answers without having to worry about "carrying the one".

Example: $$18397238$$ Partitions into $$\underline{18}\,\underline{397}\,\underline{2}\,\underline{38}.$$ Doubling each one yields $$\underline{36}\,\underline{794}\,\underline{4}\,\underline{76}$$

This is efficient if there are a lot of numbers less than $5$ in your number, but not so much otherwise.

Edit: It should be noted that this technique can work in tandem with many of the other methods mentioned, and can actually make them easier to use. For example, if you first think of the number grouped according to my strategy, then use the left-to-right method outlined in David K's answer on each group, every digit has a carried 1 except the last, so you don't even have to think about what the digit to the right is when you are doubling.

You can very well work from left to right (which is the way you spell and utter the number). Just double the current digit and anticipate a carry from the next one (if $5$ to $9$).

Examples:

$192$ yields $284$ without carries and $\bar384$ with.

$9999$ yields $08888/\bar1\bar9\bar9\bar98$.

$123456789$ yields $246802468/246\bar9\bar1\bar3\bar5\bar78$.

I don't mean that you should double all digits then process the carries. On the opposite, generate double+carry form left to right, in a single pass.

This essentially amounts to mechanically applying the regular table below: $$ 00,01,02,03,04\to0\\ 05,06,07,08,09\to1\\ 10,11,12,13,14\to2\\ 15,16,17,18,19\to3\\ 20,21,22,23,24\to4\\ 25,26,27,28,29\to5\\ 30,31,32,33,34\to6\\ 35,36,37,38,39\to7\\ 40,41,42,43,44\to8\\ 45,46,47,48,49\to9\\ 50,51,52,53,54\to0\\ 55,56,57,58,59\to1\\ 60,61,62,63,64\to2\\ 65,66,67,68,69\to3\\ 70,71,72,73,74\to4\\ 75,76,77,78,79\to5\\ 80,81,82,83,84\to6\\ 85,86,87,88,89\to7\\ 90,91,92,93,94\to8\\ 95,96,97,98,99\to9\\ $$