Audioengine HD3 vs. Sonos

Pioneer  SP-BS22-LR Speakers ($129 designed by Andrew Jones)
Musical Fidelity V-90amp ($140)
Google Chromecast Audio ($25 on sale)

You can use the optical out of the chromecast into the V-90amp.  For a little more oomph, you could add a Pioneer SW-8MK2, which is also designed by Andrew Jones of Elac Fame.  You can find it on sale for $129 occasionally at Amazon or $159 retail.  Or, drop the Pioneer Sp-BS22-LR's and add a pair of Elac B5's for $229 and you can just squeak in under budget.  

I'm not a fan of Sonos, but that's my opinion and I've installed a lot.  The App is easy to use.  I own both a Dot and an Echo.  The Echo is just OK for background listening.  I have a Squeezebox Boom sitting right next to the Echo in the Kitchen and prefer it to the Echo.  Of course, it's bigger, has stereo speakers and you can't talk to it.  

I own a pair of AudioEngine A2+'s which I like a lot for the size.  They make great computer speakers.  The HD3's are basically A2's with the addition of wireless.

In my opinion, the Chromecast's audio quality is better than the Dot's.  However, again you can use voice to interact with the Dot.  The Chromecast requires you to have the BubbleUpnp App to Cast music stored on your phone.

In my Workshop I use a Chromecast Audio driving a MF V-90AMP and a Pair of Martin logan Passage in-walls. 

Hope I didn't overwhelm you with options.  It's all subjective and you might find happiness with a pair of Play:1's or a Dot plugged into a pair of HD3's.  Good luck and keep us posted.

Another option is  a pair of Emotiva Airmotiv 5S.  Self powered with a ribbon tweeter.  I've never heard them but the ribbon tweeter has piqued my interest.  

It depends on what your looking for: ease of use or sound quality.  

They both have similar driver compliments: 3/4" tweeter and a 3"-ish mid-woofer.  They are both about the same size.  The Sonos App is really easy to use.  The HD3's have slightly better sound quality.  Remember, these are small speakers and your not going to get thumping bass out of them.  I think a pair of Play:1's will play slightly louder than the HD3's.  I've never heard much stereo imaging out of any Sonos speaker product and I've paired 1's, 3's and 5's.  

If you go the Sonos route, then I would recommend getting a Sonos Boost.  While you can use WiFi to connect the Play:1's to your Router, I've always found that using a Boost is more stable.  If your Router is in the same room as the 1's then hardwire one speaker to it if possible.  

If you're looking for a richer, more room filling sound, I would go with a Play:5 over a pair of Play:1's.  The Play:5 has more bass and will play a little louder.

Remember, you're asking for advice on an Audiophile sight.  And, for the most part, most of us here, are more concerned about sound quality over ease of use.  

BTW - While I'm writing this I'm listening to Sting on a heavily modified Squeezebox Touch.  Amplification is a Wadia 151PowerDAC Mini which is driving a pair Gallo Reference Strada's with a custom subwoofer and the sound is oh so glorious.  This is my main computer/gaming rig and I spend a lot of time in this room.

Once again, these are my opinions and yours might differ.  If possible, auditioning before buying is recommended.

Another suggestion for a stand alone speaker is Oppo Sonica at $299.  It has an aux input and you could use your Dot with it.