So my client has strictly forbidden to do any customization in SharePoint online 0365 due to the reason that future upgrades won't break. Can any one guide me what type of customizations are allowed in O365 which won't break future upgrades?
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The easiest answer on your question is probably something down the line of "What you can do within the browser is unlikely break with the next update".
General rules used:
1. No custom master pages (the new look doesn't use these)
2. Custom branding do you really need it? The theming like solutions are qutrie often a more relaiable option.
So even if you say we don't want things to break I would recommend to go down the development route!
If you look at the app model ( an interface that talks to SharePoint but isn't SharePoint) then you are at quite a low risk.