What are the reasons fields cannot be added to a hosted feature layer on the ArcGIS Online Data tab?

Answer

Hosted feature layers are the only layers in ArcGIS Online that support adding fields to a layer's attribute table. However, there are some instances where it is not possible to do so. The option to add a field to a hosted feature layer is not displayed in the Table view or the Fields view of the Data tab in ArcGIS Online, as shown in the examples below.

The following are the possible reasons for why fields cannot be added to a hosted feature layer:

Roles and privileges

In ArcGIS Online, only the owner of the hosted feature layer or the administrator of the organization is allowed to make changes to the attribute data of the layer. Non-owners without the administrator role can only view the attribute table. Refer to ArcGIS Online: Attribute field management for more information.

Similarly, only the owner of the layer or the administrator of the organization can allow other members to edit the layer. Refer to ArcGIS Online: Manage hosted feature layers for the steps to do this.

Hosted feature layers with joined attributes

A hosted feature layer with joins cannot have its attributes modified. A workaround is to add the new field before using the Join Features tool. Alternatively, create a relationship class in ArcGIS Pro using the Create Relationship Class tool instead of using a join.

Hosted feature layers with dependent layers

This is a known limitation. Creating a dependent layer, such as a hosted tile layer from a hosted feature layer creates a relationship between the two layers. This disables the option to add fields to the attribute table of the hosted feature layer. Refer to ArcGIS Online: Add or delete a field for more information.

As a workaround, delete all dependent layers to enable the option to add a field to the hosted feature layer. Refer to ArcGIS Online: Manage content for more information.

Note:
It is important to note that fields can only be added to the attribute table of hosted feature layers. Other feature layer types such as streaming feature layers or hosted feature layer views do not provide this option. Refer to ArcGIS Online: Feature layers for more information. 
A feature collection can be published as a hosted feature layer. Refer to ArcGIS Online: Publish hosted feature layers for the steps to do this.