Consider the following sequence of random variables:
$X_1$ has only values $0$ and $1$ with positive probability
$X_2$ only $0,1,2$
$X_3$ only $0,1,2,3$.
Let's stop here. Can this sequence be a martingale?
We would need to find out what $\mathbb{E}(X_2|X_1)$ is.
But I do not see how we could say anything about this without knowing the distribution of those variables.
I think you only need to say whether it can be a martingale. Obviously, you do not know enough information to say it is a martingale.
In this situation you should provide an example to show that it can be a martingale, or present a proof that it cannot possibly be one.
So:
For $n \geq 1$, Let $X_{n+1} = 0$ if $X_n = 0$ with probability $1^*$.
Otherwise, let $X_{n+1} = X_n - 1$ with probability $1/3$, $X_n$ with probability $1/3$ and $X_n+1$ with probability $1/3$.
Then $\{X_n\}$ is a martingale.
[* You might ask "Why can we do this: the question states that $X_n$ has values $0, \dots, n$ with positive probabilities?" The answer is, we can still make it such that the marginal distribution has a positive probability for each state. It is my interpretation that this is allowed. If the conditional distributions must also have positive probabilities for all states then the answer is no, it cannot be a martingale.]