The batting average for $30$ innings of a cricket player is $40$ runs. His highest score exceeds his lowest score by $100$ runs. If these two innings are not included, the average of the remaining $28$ innings is $38$ runs. Then the lowest score of the player is:

Let the highest score be $H$ and the lowest score be $L$.
From the question:

$$H-L=100\text{— (Equation 1)}$$

Total runs scored in $30$ innings:

$$\text{Total runs (30 innings)}=30\times 40=1200\text{ runs}$$

Total runs scored in $28$ innings (excluding the highest and lowest scores):

$$\text{Total runs (28 innings)}=28\times 38=1064\text{ runs}$$

The sum of the highest and lowest scores is the difference between these two totals:

$$H+L=1200-1064=136\text{— (Equation 2)}$$

Now, subtract Equation 1 from Equation 2:

$$(H+L)-(H-L)=136-100$$

$$2L=36$$

$$L=18$$

Therefore, the lowest score of the player is $18$.