Maximum item in data stream
In the streaming model, any constant approximation of the maximum requires $\Omega(N)$ space. Showing this for a small enough constant approximation follows by an easy reduction from the disjointness problem in two-party communication complexity. This is now very standard, but I will describe it below anyways.

Recall that in the disjointness problem Alice is given a binary string $x \in \{0, 1\}^N$, and Bob is given a string $y \in \{0, 1\}^N$, and they want to decide whether $x$ and $y$ have disjoint support. It is a classic result that this task requires that Alice and Bob exchange at least $\Omega(N)$ bits, even if they share random bits and only want to succeed with probability $2/3$ over the choice of randomness. 

Assume there is a randomized streaming algorithm $A$ with space complexity $S$, which with probability $2/3$ computes a number $a$ such that $\frac{a}{F_\infty} \in (\frac{2}{3}, \frac{4}{3})$, where $F_\infty$ is the true maximum frequency of the $N$ elements. This gives a protocol for the disjointness problem with $S$ bits of communication, and therefore $S = \Omega(N)$. In the reduction, Alice simulates the algorithm $A$ and feeds it a stream which increases the frequency of element $i$ by $1$ for each $i$ such that $x_i = 1$. Then Alice takes the memory state of $A$ (which has only $S$ bits) and sends it to Bob, who then simulates $A$ starting from the state he received from Alice and feeds it a stream which increases the frequency of $i$ by $1$ for each $i$ s.t. $y_i = 1$. Then Bob decides that $x$ and $y$ have disjoint support if the output $a$ of $A$ is at most $4/3$. 

For more information, see Alon, Matias, and Szegedy's beautiful paper, which introduced the streaming model and proved the first lower bounds.