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I have a series of polygons (4,000) and a series of points that exist on the perimeter of each polygon (3 to 5). The points refer to the elevation value which has previously been extracted from a DEM raster. The points exist at the minimum bounding locations of the polygons, whereby the Minimum Bounding Geometry Tool has been used to identify the outline of the polygons in a rectangular fashion and where each polygon intercepts its minimum bounding rectangle a point exists.
The polygons do not touch anywhere. An example image of the points and polygons is shown below:

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I wish to identify the point on the perimeter of each polygon that has the greatest elevation value. Any ideas how to do this in ArcMap 10.4?

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    You need to edit your question and add more information. Are these polygons overlapping, tessellating or completely individual. Are your points snapped to boundary of polygon? An image would be helpful. – Hornbydd Sep 4 '18 at 19:18
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You should use the Spatial Join tool (Analysis Tools Toolbox -> Overlay Toolset -> Spatial Join tool). Assuming the elevation values are numeric you can use the Maximum summary function in a One-To-One Spatial Join. The polygons will be the target layer and the points will be the join layer for the tool. You can change the property of the Elevation field from the default summary type (First) to be Maximum by right clicking on the field and editing the Merge Rule under the properties of the field.

You can optionally change the name of the output field to be MAX_ELEVATION if you desire by right clicking on the field and editing the Name property of the field. The output table will list the total count of points that intersect with the polygons in the Join_Count field added by the tool. The original ObjectID of the polygons will be in the Target_FID field added by the tool, so you can join the output to your original polygons and use the field calculator to transfer any data you like, if you desire.

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