Try using "Paste Special"
Notepad++ has an option, [Edit, Paste Special, Paste HTML Content]. Copying from various browsers gives different results. I've copied the same section from your question in each browser, and pasted it into Notepad++.
From Firefox:
Version:0.9
StartHTML:00000177
EndHTML:00000550
StartFragment:00000211
EndFragment:00000514
SourceURL:http://superuser.com/questions/570820/paste-rich-text-with-formating
<html><body>
<!--StartFragment-->For example, if I copied text from a webpage that read "Stackexchange" and was a clickable link to "<a href="http://stackexchange.com">http://stackexchange.com</a>",
pasting to wordpad or some office program, will paste as clickable
text. It would be great to be able to paste directly in notepad as<!--EndFragment-->
</body>
</html>
From Chrome:
Version:0.9
StartHTML:0000000185
EndHTML:0000002191
StartFragment:0000000221
EndFragment:0000002155
SourceURL:http://superuser.com/questions/570820/paste-rich-text-with-formating
<html>
<body>
<!--StartFragment--><span style="color: rgb(0, 0, 0); font-family: Arial, 'Liberation Sans', 'DejaVu Sans', sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 18px; orphans: auto; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">For example, if I copied text from a webpage that read "Stackexchange" and was a clickable link to "</span><a href="http://stackexchange.com/" style="margin: 0px; padding: 0px; border: 0px; font-size: 14px; vertical-align: baseline; background-color: rgb(255, 255, 255); color: rgb(19, 179, 189); text-decoration: none; cursor: pointer; font-family: Arial, 'Liberation Sans', 'DejaVu Sans', sans-serif; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 18px; orphans: auto; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-position: initial initial; background-repeat: initial initial;">http://stackexchange.com</a><span style="color: rgb(0, 0, 0); font-family: Arial, 'Liberation Sans', 'DejaVu Sans', sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 18px; orphans: auto; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); display: inline !important; float: none;">", pasting to wordpad or some office program, will paste as clickable text. It would be great to be able to paste directly in notepad as</span><!--EndFragment-->
</body>
</html>
From IE:
Version:1.0
StartHTML:000000231
EndHTML:000005122
StartFragment:000004682
EndFragment:000004996
StartSelection:000004682
EndSelection:000004996
SourceURL:http://superuser.com/questions/570820/paste-rich-text-with-formating
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<DIV class=post-text itemprop="description"><!--StartFragment--><P>For example, if I copied text from a webpage that read "Stackexchange" and
was a clickable link to "<A
href="http://stackexchange.com">http://stackexchange.com</A>", pasting to
wordpad or some office program, will paste as clickable text. It would be great
to be able to paste directly in notepad as</P><!--EndFragment--></DIV>
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Best Answer
This is a simple clipboard viewer that has the ability to save/load the complete state of the clipboard: http://www.freeclipboardviewer.com/
The Windows clipboard has the ability to store multiple alternative formats for data, e.g. if you copy from a web browser, you might get plain text, rich text and HTML formats stored. Many applications will also store their "native" representation of the data.
Note that for some applications (e.g. Microsoft Excel) the "native" format is actually just a reference to the source, so saving, rebooting and loading won't work quite as expected.