diff --git a/docs/source/reference-core.rst b/docs/source/reference-core.rst
index 980a3106e5..30994e4386 100644
--- a/docs/source/reference-core.rst
+++ b/docs/source/reference-core.rst
@@ -922,9 +922,6 @@ The nursery API
 
    See :meth:`~Nursery.start`.
 
-.. autoclass:: TaskStatus
-   :members:
-
 .. _task-local-storage:
 
 Task-local storage
diff --git a/docs/source/reference-lowlevel.rst b/docs/source/reference-lowlevel.rst
index bacebff5ad..faf07268cb 100644
--- a/docs/source/reference-lowlevel.rst
+++ b/docs/source/reference-lowlevel.rst
@@ -534,6 +534,8 @@ Task API
       putting a task to sleep and then waking it up again. (See
       :func:`wait_task_rescheduled` for details.)
 
+.. autoclass:: TaskStatus
+   :members:
 
 .. _guest-mode:
 
diff --git a/trio/__init__.py b/trio/__init__.py
index be7de42cde..277baa5339 100644
--- a/trio/__init__.py
+++ b/trio/__init__.py
@@ -35,7 +35,6 @@
     EndOfChannel as EndOfChannel,
     Nursery as Nursery,
     RunFinishedError as RunFinishedError,
-    TaskStatus as TaskStatus,
     TrioInternalError as TrioInternalError,
     WouldBlock as WouldBlock,
     current_effective_deadline as current_effective_deadline,
diff --git a/trio/_dtls.py b/trio/_dtls.py
index 8675cb75b6..b3ed0fd883 100644
--- a/trio/_dtls.py
+++ b/trio/_dtls.py
@@ -42,8 +42,8 @@
     from OpenSSL.SSL import Context
     from typing_extensions import Self, TypeAlias
 
-    from ._core._run import TaskStatus
-    from ._socket import Address, _SocketType
+    from trio.lowlevel import TaskStatus
+    from trio.socket import Address, _SocketType
 
 MAX_UDP_PACKET_SIZE = 65527
 
diff --git a/trio/_highlevel_open_tcp_listeners.py b/trio/_highlevel_open_tcp_listeners.py
index 6211917254..0d5f630495 100644
--- a/trio/_highlevel_open_tcp_listeners.py
+++ b/trio/_highlevel_open_tcp_listeners.py
@@ -1,8 +1,12 @@
+from __future__ import annotations
+
 import errno
 import sys
+from collections.abc import Awaitable, Callable
 from math import inf
 
 import trio
+from trio.lowlevel import TaskStatus
 
 from . import socket as tsocket
 
@@ -23,7 +27,7 @@
 # backpressure. If a connection gets stuck waiting in the backlog queue, then
 # from the peer's point of view the connection succeeded but then their
 # send/recv will stall until we get to it, possibly for a long time. OTOH if
-# there isn't room in the backlog queue... then their connect stalls, possibly
+# there isn't room in the backlog queue, then their connect stalls, possibly
 # for a long time, which is pretty much the same thing.
 #
 # A large backlog can also use a bit more kernel memory, but this seems fairly
@@ -37,16 +41,24 @@
 # so this is unnecessary -- we can just pass in "infinity" and get the maximum
 # that way. (Verified on Windows, Linux, macOS using
 # notes-to-self/measure-listen-backlog.py)
-def _compute_backlog(backlog):
-    if backlog is None:
-        backlog = inf
+def _compute_backlog(backlog: int | float | None) -> int:
     # Many systems (Linux, BSDs, ...) store the backlog in a uint16 and are
     # missing overflow protection, so we apply our own overflow protection.
     # https://github.com/golang/go/issues/5030
+    if isinstance(backlog, float):
+        # TODO: Remove when removing infinity support
+        # https://github.com/python-trio/trio/pull/2724#discussion_r1278541729
+        if backlog != inf:
+            raise ValueError(f"Only accepts infinity, not {backlog!r}")
+        backlog = None
+    if backlog is None:
+        return 0xFFFF
     return min(backlog, 0xFFFF)
 
 
-async def open_tcp_listeners(port, *, host=None, backlog=None):
+async def open_tcp_listeners(
+    port: int, *, host: str | bytes | None = None, backlog: int | float | None = None
+) -> list[trio.SocketListener]:
     """Create :class:`SocketListener` objects to listen for TCP connections.
 
     Args:
@@ -62,7 +74,7 @@ async def open_tcp_listeners(port, *, host=None, backlog=None):
           :func:`open_tcp_listeners` will bind to both the IPv4 wildcard
           address (``0.0.0.0``) and also the IPv6 wildcard address (``::``).
 
-      host (str, bytes-like, or None): The local interface to bind to. This is
+      host (str, bytes, or None): The local interface to bind to. This is
           passed to :func:`~socket.getaddrinfo` with the ``AI_PASSIVE`` flag
           set.
 
@@ -78,13 +90,16 @@ async def open_tcp_listeners(port, *, host=None, backlog=None):
           all interfaces, pass the family-specific wildcard address:
           ``"0.0.0.0"`` for IPv4-only and ``"::"`` for IPv6-only.
 
-      backlog (int or None): The listen backlog to use. If you leave this as
-          ``None`` then Trio will pick a good default. (Currently: whatever
+      backlog (int, math.inf, or None): The listen backlog to use. If you leave this as
+          ``None`` or ``math.inf`` then Trio will pick a good default. (Currently: whatever
           your system has configured as the maximum backlog.)
 
     Returns:
       list of :class:`SocketListener`
 
+    Raises:
+      :class:`TypeError` if invalid arguments.
+
     """
     # getaddrinfo sometimes allows port=None, sometimes not (depending on
     # whether host=None). And on some systems it treats "" as 0, others it
@@ -93,7 +108,7 @@ async def open_tcp_listeners(port, *, host=None, backlog=None):
     if not isinstance(port, int):
         raise TypeError(f"port must be an int not {port!r}")
 
-    backlog = _compute_backlog(backlog)
+    computed_backlog = _compute_backlog(backlog)
 
     addresses = await tsocket.getaddrinfo(
         host, port, type=tsocket.SOCK_STREAM, flags=tsocket.AI_PASSIVE
@@ -126,7 +141,7 @@ async def open_tcp_listeners(port, *, host=None, backlog=None):
                     sock.setsockopt(tsocket.IPPROTO_IPV6, tsocket.IPV6_V6ONLY, 1)
 
                 await sock.bind(sockaddr)
-                sock.listen(backlog)
+                sock.listen(computed_backlog)
 
                 listeners.append(trio.SocketListener(sock))
             except:
@@ -150,14 +165,14 @@ async def open_tcp_listeners(port, *, host=None, backlog=None):
 
 
 async def serve_tcp(
-    handler,
-    port,
+    handler: Callable[[trio.SocketStream], Awaitable[object]],
+    port: int,
     *,
-    host=None,
-    backlog=None,
-    handler_nursery=None,
-    task_status=trio.TASK_STATUS_IGNORED,
-):
+    host: str | bytes | None = None,
+    backlog: int | float | None = None,
+    handler_nursery: trio.Nursery | None = None,
+    task_status: TaskStatus = trio.TASK_STATUS_IGNORED,  # type: ignore[assignment]  # default has type "_TaskStatusIgnored", argument has type "TaskStatus"
+) -> None:
     """Listen for incoming TCP connections, and for each one start a task
     running ``handler(stream)``.
 
diff --git a/trio/_tests/test_highlevel_open_tcp_listeners.py b/trio/_tests/test_highlevel_open_tcp_listeners.py
index e58cbd13cc..6eca844f0c 100644
--- a/trio/_tests/test_highlevel_open_tcp_listeners.py
+++ b/trio/_tests/test_highlevel_open_tcp_listeners.py
@@ -1,6 +1,7 @@
 import errno
 import socket as stdlib_socket
 import sys
+from math import inf
 
 import attr
 import pytest
@@ -289,6 +290,7 @@ async def test_open_tcp_listeners_backlog():
     tsocket.set_custom_socket_factory(fsf)
     for given, expected in [
         (None, 0xFFFF),
+        (inf, 0xFFFF),
         (99999999, 0xFFFF),
         (10, 10),
         (1, 1),
@@ -297,3 +299,13 @@ async def test_open_tcp_listeners_backlog():
         assert listeners
         for listener in listeners:
             assert listener.socket.backlog == expected
+
+
+async def test_open_tcp_listeners_backlog_float_error():
+    fsf = FakeSocketFactory(99)
+    tsocket.set_custom_socket_factory(fsf)
+    for should_fail in (0.0, 2.18, 3.14, 9.75):
+        with pytest.raises(
+            ValueError, match=f"Only accepts infinity, not {should_fail!r}"
+        ):
+            await open_tcp_listeners(0, backlog=should_fail)
diff --git a/trio/_tests/verify_types.json b/trio/_tests/verify_types.json
index ac2cfbd197..9c71ddd58a 100644
--- a/trio/_tests/verify_types.json
+++ b/trio/_tests/verify_types.json
@@ -7,11 +7,11 @@
     "warningCount": 0
   },
   "typeCompleteness": {
-    "completenessScore": 0.9154704944178629,
+    "completenessScore": 0.9170653907496013,
     "exportedSymbolCounts": {
       "withAmbiguousType": 0,
-      "withKnownType": 574,
-      "withUnknownType": 53
+      "withKnownType": 575,
+      "withUnknownType": 52
     },
     "ignoreUnknownTypesFromImports": true,
     "missingClassDocStringCount": 1,
@@ -108,7 +108,6 @@
       "trio.lowlevel.wait_writable",
       "trio.open_ssl_over_tcp_listeners",
       "trio.open_ssl_over_tcp_stream",
-      "trio.open_tcp_listeners",
       "trio.open_unix_socket",
       "trio.run",
       "trio.run_process",
diff --git a/trio/lowlevel.py b/trio/lowlevel.py
index 36d23d5955..c66e22b60e 100644
--- a/trio/lowlevel.py
+++ b/trio/lowlevel.py
@@ -15,6 +15,7 @@
     RaiseCancelT as RaiseCancelT,
     RunVar as RunVar,
     Task as Task,
+    TaskStatus as TaskStatus,
     TrioToken as TrioToken,
     UnboundedQueue as UnboundedQueue,
     UnboundedQueueStatistics as UnboundedQueueStatistics,