The best you can do is...
1. Activate the Spot Elevation tool.
2. In the Properties pallet, click Edit Type.
3. In the Type Properties dialog box, click Duplicate.
4. In the Type Properties dialog box, click Rename.
5. Add the text NO ARROW into the name.
6. In the Type Properties dialog box, set the Leader Arrowhead parameter to the value None.
7. Click Okay to exit the Type Properties dialog box.
Now when you place a Spot Elevation marker, it takes three clicks. Once to place the actual elevation source point, a second time to place a kink in the leader, and a third time to place the marker itself. A leader line will not be created if you click all three times in the same place...in other words, you want the marker to be at the source point. It only gets created if you put the marker someplace other than the source point.
The arrow will show regardless, which is what the above seven steps changes.
So now you can activate the tool, choose your new "no arrow" type in the properties pallet, and click three times at your point to place a leaderless marker.