We can use Zorn's lemma to prove that for a non-zero ring $R$ with left(right) identity, every proper right(left) ideal of $R$ is contained in a maximal right(left) ideal. The existence of left (right) identity is important.
But we can't use the same method to prove the existence of a maximal left ideal.
There is a lot of examples of non-zero commutative rings without identity which don't have a maximal ideal. For example, $(\mathbb{Q},+)$ doesn't have a maximal subgroup. Then let $xy=0$ for all $x,y\in\mathbb{Q}$.
But I don't know if there is a ring with left identity but doesn't have any maximal left ideal.
Best Answer
I think there is not, owing to the symmetry of the Jacobson radical in rings without identity.
Jacobson developed a characterization of (what is now called) the Jacobson radical as the intersection of all regular right maximal ideals, equivalent to the intersection of all regular maximal left ideals.
Here “regular” applied to a right ideal $T$ means “there exists an element $e$ such that $ex-x\in T$ for all $x\in R$”. Clearly a left identity would suffice for making all right ideals regular. The left hand counterpart is apparent.
So if the ring has a left identity, its Jacobson radical is proper, indicating that there must exist some maximal (proper, of course) regular left ideal in $R$ too.