Think of a number (call it 'c'). Square it, and then add it to the square. This gives you a new number. Continue by squaring this number, and adding your original number 'c'. Repeat to construct a sequence of numbers.
Starting with 3, you square it (9) and add 3 to get 12. Then square that (144) and add 3 to get 147. You get a sequence of numbers, 3, 12, 147, 21612, … which is clearly growing unboundedly.
Starting with 2, you get this sequence instead: 2, 6, 38, 1446, … which is also growing unboundedly.
Starting with 1, you get 1, 2, 5, 26, … which is again growing unboundedly.
Does it always grow unboundedly?
Starting with 0, you get 0, 0, 0, 0, … so no, it doesn't always.
Starting with -1, you get -1, 0, -1, 0, … which again isn't growing.
Starting with -2, you get -2, 2, 2, 2, … which isn't growing.
If you try decimals between -2 and 0.25, they also don't grow. If you try 0.26 it does grow, but much more slowly at first.
We could colour the number line solid black between -2 and 0.25, to show this region where it doesn't grow. We could then say “if it grows bigger than 10 in one step, colour it red; if it grows bigger than 10 in two steps, colour it lighter red; if it grows bigger than 10 in three steps, colour it even lighter red, …, if it grows bigger than 10 in 25 steps colour it yellow,” and so on, with a whole range of colours. We'd end up with a number line coloured like this:
Below is a zoom of the region 0.25 to 0.27, the colours indicating increasingly slow divergence rates closer to 0.25.
The colours cycle through 100 shades: 25 shades of red through to yellow, 25 shades of yellow through to dark grey, 25 shades of dark grey through to white, and finally 25 shades from white through to red again. This means when red appears again after white, near 0.25, the sequence is taking over 100 terms to get above 10. You can see a very small band of red just about above the '2' of 0.25, which means the colour cycle has gone round twice, and the sequences are taking over 200 terms to get above 10.