What are some great profitable types of services someone can start a consultancy service with solo, and later grow?

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Wow - what an interesting question. Have you considered sustainable services? Services that will endure no matter what the economy is doing or what the state of things is. Healthcare, medical services, education, financials, etc? Or really honing in on a niche — marketing/advertising when budgets are cut? Business strategy while the stock market is down? Things like that would definitely pique the interest of businesses. Define your ideal customer, one who understands that there will be a payment involved. And you have control over when you get paid and your payment structure. You can do a discovery conversation upfront for no charge, but set a time limit. Create a discovery questionnaire as an initital qualifier to make sure a potential client is a good fit for you before you and the client highly invest your time. I hope that helps!

So the types of services you offer or considering are on demand and you can charge thousands of dollars for what you have in mind. When it comes to something you can grow over time that is any business. It all depends how much time and effort your willing to put in to your business. But as you stated its over time, don't quit your day job. As your starting up this business donit in the side and on your spare time. Then as business picks up little by little you can gradually begin considering doing your business full time. What you are asking are things that I do, I am a career consultant as well as business start up and entrepreneur consultant. Like I said it all depends on the effort and time you put in specifically on the things that brings a high return like, cold calls, contacting businesses, individuals that need the service you providing. And researching and contacting the decision makers in your target market.
The first 3 you mentioned are very profitable specifically web development, then video editing and instagram marketing in that order. From most profitable to least. And last nugget... it also depends on how you position your self as an expert in that market. And that takes a bit of time, personal development, and passion but its 100% possible. In bbn other words simply do the things you have to do to get your name up there. You should be the first person people think about when they ask for your service. I hope this helps enjoy.