I confirm @isaacg's results. I found some more sequences of length 6 that may be useful for spotting a pattern:
11116657 = 29 * 383333.
11116658 = 2 * 7 * 794047.
11116659 = 23 * 483333.
11116660 = 5 * 524 * 4243.
11116661 = 43 * 258527.
11116662 = 3 * 3705554.
11666666 = 22 * 530303.
11666667 = 3 * 3888889.
11666668 = 52 * 224359.
11666669 = 47 * 248227.
11666670 = 2355 * 4954.
11666671 = 29 * 402299.
64664456 = 2 * 32332228.
64664457 = 33 * 923 * 2123.
64664458 = 2 * 32332229.
64664459 = 173 * 373783.
64664460 = 27 * 28 * 85535.
64664461 = 29 * 2229809.
161116659 = 33 * 4882323.
161116660 = 5 * 32223332.
161116661 = 53 * 3039937.
161116662 = 3 * 53705554.
161116663 = 29 * 5555747.
161116664 = 2 * 80558332.
2221828216 = 4 * 555457054.
2221828217 = 49 * 45343433.
2221828218 = 3 * 740609406.
2221828219 = 73 * 30436003.
2221828220 = 5 * 444365644.
2221828221 = 3 * 740609407.
Also I believe there are infinitely many quadruples!
1909899 = 3 * 636633.
1909900 = 4 * 477475.
1909901 = 7 * 272843.
1909902 = 3 * 636634.
Now insert "09".
190909899 = 3 * 63636633.
190909900 = 4 * 47727475.
190909901 = 7 * 27272843.
190909902 = 3 * 63636634.
We can continue inserting "09" indefinitely. So we have 19(09)899 - 19(09)902 where (09) is repeated 1 or more times is always a quadruple of consecutive Melissa numbers.