node.js prevents the stack overgrowth you describe by using asynchronous techniques everywhere1.
Anything that could block uses callbacks for further processing, not blocking calls. This avoids stack growth completely, and makes it easy to re-enter the event loop (that "drives" the underlying real I/O and request dispatching).
Consider this pseudo-code:
fun() {
  string = net.read();
  processing(string);
}
Thread is blocked on read, stack can only be free'd up after both the read completes, and processing is done.
Now if all your code is like:
fun() {
  net.read(onDone: processing(read_data));
}
And if you implement read like this:
net.read(callback) {
  iorequest = { read, callback };
  io.push_back(iorequest);
}
fun is done as soon as read can queue a read I/O with the associated callback. fun's stack is rewound without blocking - it returns "immediately" to the event loop without any thread stack leftovers.
I.e. you can move on to the next callback (re-enter the event loop) without keeping any per-request data on the thread stack.
So node.js avoid stack overgrowth by using asynchronous callbacks wherever blocking calls would happen in "user" code.
For more about this, please check out the node.js 'about' page, and the first set of slides linked at the end. 
1well, nearly I guess 
You mention QueueUserAPC in a comment. With that type of processing, a queued APC is allowed to block, and the next APC in the queue gets processed on the thread's stack, making it a "recursive" dispatch.
Say we have three APCs pending (A, B and C). We get:
Initial state:
Queue   ABC
Stack   xxxxxxxx
Thread sleeps so APC dispatch starts, enters processing for A:
Queue   BC
Stack   AAAAxxxxxxxx
A blocks, B is dispatched on the same stack:
Queue   C
Stack   BBBBBBAAAAxxxxxxxx
B blocks, C is dispatched:
Queue   
Stack   CCCCCCCBBBBBBAAAAxxxxxxxx
It's clearly visible that if enough blocking APCs are pending, the stack will eventually blow up.
With node.js, the requests are not allowed to block. Instead, here's a mock-up of what would happen for the same three requests:
Queue      ABC
Stack      xxxxxxxx
A starts processing:
Queue      BC
Stack      AAAAxxxxxxxx
Now A needs to do something that blocks - in node.js, it actually can't. What it does is queue another request (A') (presumably with a context - simplistically a hash with all your variables):
I/O queue  A'
Queue      BC
Stack      AAAAxxxxxxxx
Then A returns and were's back to:
I/O queue  A'
Queue      BC
Stack      xxxxxxxx
Notice: no more A stackframe. The I/O pending queue is actually managed by the OS (using epoll or kqueue or whatever). The main thread checks both the OS I/O ready states and pending (needing CPU) queues in the event loop.
Then B gets some CPU:
I/O queue  A'
Queue      C
Stack      BBBBBBBxxxxxxxx
Same thing, B wants to do I/O. It queues a new callback and returns.
I/O queue  A'B'
Queue      C
Stack      xxxxxxxx
If B's I/O request completes in the mean time, the next snapshot could look like
I/O queue  A'
Queue      B'
Stack      CCCCCxxxxxxxx
At no point is there more than one callback stack frame on the processing thread. Blocking calls are not provided by the API, that stack doesn't exhibit the type of recursive growth the APC pattern does.