Let $k$ be an algebraically closed field. A variety over $k$ is a separated integral scheme of finite type over $k$. Let $V$ be a complete non-projective non-singular variety over $k$. Let $Z$ be a closed subset of $V$. Let $\mathcal I$ be the ideal sheaf which defines $Z$ as a reduced closed subscheme of $V$. $\mathcal I^n$ defines a closed subscheme $Z_n$ of $V$ for every integer $n \ge 1$. I would like to know examples of closed subschemes of $V$ other than $Z_n$.
Remark The more examples, the better. Please don't think that the question would be solved if one example would be given.
Edit(March 23, 2014) I have just posted a similar question in MathOverflow.
Take the subscheme consisting of a double point and a simple point !
Edit
As requested in the comments, here are a few more details.
Choose two closed points $x,y\in V$.
Let $\mathcal I(x)\subsetneq \mathcal O_V$ and $\mathcal I(y)\subsetneq \mathcal O_V$ be the ideal sheaves of these two points considered as reduced closed points of $V$.
Then define the ideal sheaf $\mathcal K\subset \mathcal O_V$ to be the unique sheaf of ideals of $\mathcal O_V$ satisfying:
a) $\mathcal K|(V\setminus \{y\})=\mathcal I(x)|(V\setminus \{y\})$
b) $\mathcal K|(V\setminus \{x\})=\mathcal I(y)^2|(V\setminus \{x\})$
If $\mathcal I$ is the ideal sheaf of the reduced subscheme $Z=\{x,y\}$ we then have $$ \cdots\subsetneq \mathcal I^n\subsetneq \cdots \mathcal I^2 \subsetneq \mathcal K \subsetneq \mathcal I \subsetneq \mathcal O_V $$ so that $\mathcal K$ defines a non-reduced subscheme $Z'$ whose reduction is $Z$ and satisfying $$Z=Z_1\subsetneq Z'\subsetneq Z_2\subsetneq \cdots\subsetneq Z_n\subsetneq\cdots V$$ so that $Z'$ is distinct from the all the thickenings $Z_n$ of $Z$ defined by the $\mathcal I^n$.