Any trick to defining an enum as flags/powers of 2 without eventually needing a calculator?

Write the values as shifted bits and let the compiler do the math:

[Flags]
private enum Targets : uint
{
    None                = 0,
    Campaigns           = 1,
    CampaignGroups      = 2 << 0,
    Advertisers         = 2 << 1,
    AdvertiserGroups    = 2 << 2,
    AffiliateGroups     = 2 << 3,
    Affiliates          = 2 << 4,
    Creatives           = 2 << 5,
    DetailedLeads       = 2 << 6,
    DetailedSales       = 2 << 7,
    ProgramLeads        = 2 << 8,
    CreativeDeployments = 2 << 9,
    CampaignCategories  = 2 << 10,
    Payouts             = 2 << 11,
    // etc.
}

James’s suggestion is a good one, too. In fact I like this way even better. You could also write it like this:

[Flags]
private enum Targets : uint
{
    None                = 0,
    Campaigns           = 1 << 0,
    CampaignGroups      = 1 << 1,
    Advertisers         = 1 << 2,
    AdvertiserGroups    = 1 << 3,
    AffiliateGroups     = 1 << 4,
    // etc.
}