Shortest program that continuously allocates memory
Funge-98 (cfunge), 1 byte
9

I would have posted this earlier, but decided to test it, and it took a while to get my computer back to a usable state. cfunge stores the Funge stack on the operating system's heap (which is easily verifiable by running the program with a small memory limit, something that I should have done earlier!), so an infinitely growing stack (as with this program, which just pushes 9 repeatedly; Funge programs wrap from the end of a line back to the start by default) will allocate memory forever. This program likely also works in some Befunge-93 implementations.
More interesting:
&quot;NULL #(4

This was my first idea, and is an infinite allocation that doesn't rely on the Funge stack (although it blows up the Funge stack too). To start with, the &quot; command pushes a copy of the rest of the program to the stack (it's a string, and the program wraps round, so the close quote also serves as the open quote). Then the N reflects (it has no meaning by default), causing the program to run backwards. The &quot; runs again, and pushes the program to the stack – the other way round this time, with the N at the top of the stack – then the program wraps around, loading a library with a 4-letter name (4(; the NULL library is part of cfunge's standard library). NULL defines all uppercase letters to do reflect, so the L reflects, the # skips the library load on the way back, the 4 pushes junk we don't care about to the stack and the whole program repeats from the start. Given that loading a library multiple times has an effect, and requires the library's command list to be stored once for each copy of the library (this is implied by Funge-98's semantics), it leaks memory via non-stack storage (which is an alternative method of defining &quot;heap&quot;, relative to the language rather than the OS).