diff --git a/Cargo.lock b/Cargo.lock
index c43ba9d9c..fe05f99a2 100644
--- a/Cargo.lock
+++ b/Cargo.lock
@@ -1758,6 +1758,7 @@ dependencies = [
  "effectcraft-raster",
  "kurbo",
  "linesweeper",
+ "log",
  "tiny-skia 0.12.0",
 ]
 
diff --git a/crates/path/Cargo.toml b/crates/path/Cargo.toml
index 2e9bc85d9..122c37fe0 100644
--- a/crates/path/Cargo.toml
+++ b/crates/path/Cargo.toml
@@ -14,5 +14,6 @@ effectcraft-geom = { workspace = true }
 effectcraft-keyframe = { workspace = true }
 effectcraft-raster = { workspace = true }
 kurbo = { workspace = true }
+log = { workspace = true }
 tiny-skia = { workspace = true }
 linesweeper = "0.5"
diff --git a/crates/path/src/ops.rs b/crates/path/src/ops.rs
index 826554a5d..5c89ce6bc 100644
--- a/crates/path/src/ops.rs
+++ b/crates/path/src/ops.rs
@@ -351,6 +351,11 @@ fn round_sub(sub: &Sub, r: f64, out: &mut BezPath) {
 /// join. `copies` copies are made, copy *i* (0-based) offset by `amount · (i + 1 + copy_offset)`,
 /// all returned in the same compound path.
 pub fn offset(paths: &[BezPath], amount: f64, join: Join, miter_limit: f64, copies: f64, copy_offset: f64) -> Vec<BezPath> {
+    offset_within(paths, amount, join, miter_limit, copies, copy_offset, MAX_OPERATOR_SEGMENTS)
+}
+
+/// [`offset`] with each path's result held to `max` elements (its first copy aside).
+fn offset_within(paths: &[BezPath], amount: f64, join: Join, miter_limit: f64, copies: f64, copy_offset: f64, max: usize) -> Vec<BezPath> {
     let n = pieces_within_budget(paths, copies.round().clamp(1.0, 1000.0) as usize);
     paths
         .iter()
@@ -358,7 +363,16 @@ pub fn offset(paths: &[BezPath], amount: f64, join: Join, miter_limit: f64, copi
             let mut out = BezPath::new();
             for i in 0..n {
                 let d = amount * (i as f64 + 1.0 + copy_offset);
-                out.extend(crate::offset::offset_path(p, d, join, miter_limit).elements().iter().copied());
+                let copy = crate::offset::offset_path(p, d, join, miter_limit);
+                // The budget is spent on what `offset_path` returns (an open subpath comes back as
+                // a closed outline of many segments; a zero offset as the path itself): the first
+                // copy is always kept, and a copy that would pass the budget is not made, so the
+                // work stays within a small multiple of it.
+                if i > 0 && out.elements().len().saturating_add(copy.elements().len()) > max {
+                    log::warn!("path: Offset Paths made {i} of {n} copies; more would pass {max} segments");
+                    break;
+                }
+                out.extend(copy.elements().iter().copied());
             }
             out
         })
diff --git a/crates/path/src/ops/tests.rs b/crates/path/src/ops/tests.rs
index 2de236e0c..e64a55b47 100644
--- a/crates/path/src/ops/tests.rs
+++ b/crates/path/src/ops/tests.rs
@@ -182,6 +182,25 @@ fn offset_copies() {
     assert!(close(o[0].bounding_box().width(), 140.0, 1e-3));
 }
 
+#[test]
+fn offset_copies_are_budgeted_by_the_output_they_return() {
+    // An open one-segment subpath comes back as a closed outline of several elements, so the copies
+    // stop when the returned elements pass the budget, not when the input segments would.
+    let line = || {
+        let mut p = BezPath::new();
+        p.move_to((0.0, 0.0));
+        p.line_to((50.0, 0.0));
+        p
+    };
+    let one = offset_within(&[line()], 5.0, Join::Round, 4.0, 1.0, 0.0, usize::MAX)[0].elements().len();
+    assert!(one > 4, "{one}");
+    let o = offset_within(&[line()], 5.0, Join::Round, 4.0, 50.0, 0.0, one * 3);
+    let moves = o[0].elements().iter().filter(|e| matches!(e, PathEl::MoveTo(_))).count();
+    assert!((1..50).contains(&moves) && o[0].elements().len() <= one * 4, "{moves} contours, {} elements", o[0].elements().len());
+    // The first copy always comes back, however small the budget.
+    assert!(!offset_within(&[line()], 5.0, Join::Round, 4.0, 5.0, 0.0, 0)[0].elements().is_empty());
+}
+
 #[test]
 fn offset_open_path_outlines() {
     let mut l = BezPath::new();