In typical experimental tradition, let's write code to check this out.
(defun difficult-to-double-p (n) (loop :for (x y) := (multiple-value-list (floor n 10)) :then (multiple-value-list (floor x 10)) :thereis (> y 4) :until (zerop x))) (defun test (max-pow) (loop :for x := 1 :then (* 2 x) :for i :from 0 :to max-pow :unless (difficult-to-double-p x) :do (format t "~D " i)))
This will print out the powers of 2 that satisfy your condition. I tried it up to $n=500,000$ and only came up with $\{0, 1, 2, 5, 10\}$. Is it a coincidence that those powers of 2, except for $n=0$, are factors of 10?
Here is the output for the curious.
> (test 500000) 0 1 2 5 10 Evaluation took: 174.270 seconds of real time 174.940405 seconds of total run time (173.307653 user, 1.632752 system) [ Run times consist of 5.225 seconds GC time, and 169.716 seconds non-GC time. ] 100.38% CPU 371,799,804,896 processor cycles 46,761,694,928 bytes consed NIL
EDIT: Instead of doubling, let's try $k$-tupling. Here's a good ol' table for $2 \le n \le 100$:
2: 0 1 2 5 10 3: 0 1 5 4: 0 1 5 5: 0 6: 0 7: 0 3 4 8: 0 9: 0 10: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 11: 0 1 2 3 12: 0 1 2 13: 0 1 14: 0 1 15: 0 16: 0 17: 0 18: 0 2 6 19: 0 4 20: 0 1 2 5 10 21: 0 1 2 22: 0 1 23: 0 1 24: 0 1 25: 0 26: 0 27: 0 28: 0 29: 0 30: 0 1 5 31: 0 1 32: 0 1 2 33: 0 1 34: 0 1 35: 0 36: 0 37: 0 38: 0 2 39: 0 4 40: 0 1 5 41: 0 1 42: 0 1 43: 0 1 44: 0 1 45: 0 46: 0 47: 0 48: 0 2 49: 0 2 50: 0
EDIT 2: Next, let's try changing the base $b$, where the digits must be between $0$ and $\tfrac{b}{2}-1$ if $b$ is even, $\lfloor b/2\rfloor$ if odd. Here's a table of $b$ for $2 \le n\le 100$.
3: 0 2 8 4: 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 5: 0 1 5 8 6: 0 1 3 9 7: 0 1 3 4 6 9 8: 0 1 3 4 6 7 9 10 12 13 15 16 18 19 21 22 24 25 27 28 30 31 33 34 36 37 39 40 42 43 45 46 48 49 51 52 54 55 57 58 60 61 63 64 66 67 69 70 72 73 75 76 78 79 81 82 84 85 87 88 90 91 93 94 96 97 99 100 9: 0 1 2 8 13 14 10: 0 1 2 5 10 11: 0 1 2 4 8 14 30 12: 0 1 2 4 6 12 13: 0 1 2 4 5 9 24 14: 0 1 2 4 5 8 10 25 15: 0 1 2 4 5 6 8 9 12 13 16 24 16: 0 1 2 4 5 6 8 9 10 12 13 14 16 17 18 20 21 22 24 25 26 28 29 30 32 33 34 36 37 38 40 41 42 44 45 46 48 49 50 52 53 54 56 57 58 60 61 62 64 65 66 68 69 70 72 73 74 76 77 78 80 81 82 84 85 86 88 89 90 92 93 94 96 97 98 100 17: 0 1 2 3 11 18 18: 0 1 2 3 7 13 19: 0 1 2 3 6 14 18 19 34 20: 0 1 2 3 6 7 11 21: 0 1 2 3 6 7 9 12 18 45 22: 0 1 2 3 5 9 11 12 19 23: 0 1 2 3 5 8 24: 0 1 2 3 5 7 25 25: 0 1 2 3 5 7 8 14 21 26: 0 1 2 3 5 6 13 27: 0 1 2 3 5 6 8 13 28: 0 1 2 3 5 6 8 12 29: 0 1 2 3 5 6 7 10 30: 0 1 2 3 5 6 7 10 11 13 15 22 31: 0 1 2 3 5 6 7 8 10 11 12 15 16 17 20 21 25 36 32: 0 1 2 3 5 6 7 8 10 11 12 13 15 16 17 18 20 21 22 23 25 26 27 28 30 31 32 33 35 36 37 38 40 41 42 43 45 46 47 48 50 51 52 53 55 56 57 58 60 61 62 63 65 66 67 68 70 71 72 73 75 76 77 78 80 81 82 83 85 86 87 88 90 91 92 93 95 96 97 98 100 33: 0 1 2 3 4 14 23 42 34: 0 1 2 3 4 9 17 49 35: 0 1 2 3 4 8 12 14 40 36: 0 1 2 3 4 8 9 37: 0 1 2 3 4 7 11 38: 0 1 2 3 4 7 9 14 21 38 39: 0 1 2 3 4 7 9 13 16 33 40: 0 1 2 3 4 7 8 11 20 27 41: 0 1 2 3 4 7 8 9 15 42: 0 1 2 3 4 7 8 9 14 19 20 28 43: 0 1 2 3 4 6 12 20 44: 0 1 2 3 4 6 12 13 14 22 45: 0 1 2 3 4 6 9 12 13 14 15 24 25 26 36 45 46: 0 1 2 3 4 6 9 10 15 47: 0 1 2 3 4 6 8 19 21 39 48: 0 1 2 3 4 6 8 10 14 20 49: 0 1 2 3 4 6 8 9 13 50: 0 1 2 3 4 6 8 9 10 15 17 23
And here's a pictorial representation. Dot means easy, space means not.
4: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5: .. . . 6: .. . . 7: .. .. . . 8: .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9: ... . .. 10: ... . . 11: ... . . . . 12: ... . . . 13: ... .. . . 14: ... .. . . . 15: ... ... .. .. . . 16: ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . 17: .... . . 18: .... . . 19: .... . . .. . 20: .... .. . 21: .... .. . . . . 22: .... . . .. . 23: .... . . 24: .... . . . 25: .... . .. . . 26: .... .. . 27: .... .. . . 28: .... .. . . 29: .... ... . 30: .... ... .. . . . 31: .... .... ... ... .. . . 32: .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . 33: ..... . . . 34: ..... . . . 35: ..... . . . . 36: ..... .. 37: ..... . . 38: ..... . . . . . 39: ..... . . . . . 40: ..... .. . . . 41: ..... ... . 42: ..... ... . .. . 43: ..... . . . 44: ..... . ... . 45: ..... . . .... ... . . 46: ..... . .. . 47: ..... . . . . . 48: ..... . . . . . 49: ..... . .. . 50: ..... . ... . . . 51: ..... . ... . . . 52: ..... .. ... . 53: ..... .. . . . . 54: ..... .. . . 55: ..... .. . . 56: ..... .. .. . . 57: ..... ... . . . 58: ..... ... . 59: ..... ... . . . . . . . 60: ..... ... . . .. . 61: ..... .... .. . . 62: ..... .... ... . .. .. . 63: ..... ..... .... .... ... .. . . 64: ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... 65: ...... . . . 66: ...... . . 67: ...... . . . 68: ...... .. . . . 69: ...... . . 70: ...... . . . . . 71: ...... .. .. . . . . 72: ...... ... . . .. 73: ...... ... . . 74: ...... . . 75: ...... . . . . 76: ...... . . . . . 77: ...... . . . .. . 78: ...... . .. . . . . 79: ...... .. . .. . 80: ...... .. . .. 81: ...... .. . .. 82: ...... ... . . 83: ...... ... . .. . 84: ...... .... . . . . . 85: ...... .... . ... . . . . . 86: ...... . . . . . . 87: ...... . .. .. . 88: ...... . . ... . 89: ...... . . .... . . 90: ...... . . ..... .... .. 91: ...... . . . 92: ...... . .. . 93: ...... . ... . . . 94: ...... . . . . 95: ...... . . . . . . 96: ...... . . . . . 97: ...... . . .. . . 98: ...... . .. . 99: ...... . .. . 100: ...... . ... . . .
EDIT 3: Define base $b$ to be easy if there are an infinite number of easy doublings.
CONJECTURE 1: The only easy bases are $2^j$ for $j\ge 2$.
A friend noticed...
CONJECTURE 2: The only hard-to-double numbers in base $2^j$ are numbers whose power is $\{ij-1\mid i\ge 1\}$.
And from the picture, notice when the leftmost column gets thicker. It happens after each power of two.
CONJECTURE 3: $2^n$ in base $2^k$ is easy to double for all $k > n$.