l2 = Table[FromDigits@{i, j}, {i, 2, 9}, {j, i, 9}] // Flatten;
l3 = Table[FromDigits@{i, j, k}, {i, 2, 9}, {j, i, 9}, {k, j, 9}] // Flatten;
l4 = Table[FromDigits@{i, j, k, l}, {i, 2, 9}, {j, i, 9}, {k, j, 9}, {l, k, 9}] // Flatten;
desiredList = Flatten[{l2, l3, l4}]

f[344] = 345
 f[348] = 349
 f[349] = 355
 f[399] = 444

f[9999] = 22222